David M Rouse

Romuald Pepion
email: romuald.pepion@univ-nantes.fr

Sheila S Hemami
email: hemami@ece.cornell.edu

Patrick Le Callet
email: patrick.lecallet@univ-nantes.fr

Image Utility Assessment and a Relationship with Image Quality Assessment

Present quality assessment (QA) algorithms aim to generate scores for natural images consistent with subjective scores for the quality assessment task. For the quality assessment task, human observers evaluate a natural image based on its perceptual resemblance to a reference. Natural images communicate useful information to humans, and this paper investigates the utility assessment task, where human observers evaluate the usefulness of a natural image as a surrogate for a reference. Current QA algorithms implicitly assess utility insofar as an image that exhibits strong perceptual resemblance to a reference is also of high utility. However, a perceived quality score is not a proxy for a perceived utility score: a decrease in perceived quality may not affect the perceived utility. Two experiments are conducted to investigate the relationship between the quality assessment and utility assessment tasks. The results from these experiments provide evidence that any algorithm optimized to predict perceived quality scores cannot immediately predict perceived utility scores. Several QA algorithms are evaluated in terms of their ability to predict subjective scores for the quality and utility assessment tasks. Among the QA algorithms evaluated, the visual information fidelity (VIF) criterion, which is frequently reported to provide the highest correlation with perceived quality, predicted both perceived quality and utility scores reasonably. The consistent performance of VIF for both the tasks raised suspicions in light of the evidence from the psychophysical experiments. A thorough analysis of VIF revealed that it artificially emphasizes evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). A modified implementation of VIF, denoted VIF*, is presented that provides statistically significant improvement over VIF for the quality assessment task and statistically worse performance for the utility assessment task. A novel utility assessment algorithm, referred to as the natural image contour evaluation (NICE), is introduced that conducts a comparison of the contours of a test image to those of a reference image across multiple image scales to score the test image. NICE demonstrates a viable departure from traditional QA algorithms that incorporate energy-based approaches and is capable of predicting perceived utility scores.

INTRODUCTION

Present quality assessment ‡ (QA) algorithms aim to generate scores for natural images consistent with subjective scores for the quality assessment task. For the quality assessment task, human observers evaluate a natural image based on its perceptual resemblance to a reference. The reference may be either an explicit, external natural image or an internal reference, only accessible to the observer. Since natural images communicate useful information to humans, this paper investigates the utility assessment task. For the utility assessment task, human observers evaluate the usefulness of a natural image as a surrogate for a reference.

No algorithms exist for the utility assessment task for natural images. Driven by the quality assessment task, current QA algorithms for natural images implicitly assess utility insofar as an image that exhibits strong perceptual resemblance to the reference is also of high utility. However, applications that use other, substantially constrained classes of images (e.g., scientific and medical images) evaluate the "quality" of an image in terms of the usefulness of that image to a human observer performing a task (e.g., detecting the presence of a tumor). [START_REF] Barrett | Objective assessment of image quality IV: Application to adaptive optics[END_REF] Model observers that aim to imitate the performance of human observers for a particular task have been designed and studied for medical and scientific images. [START_REF] Barrett | Model observers for assessment of image quality[END_REF] Such model observers constitute utility assessment algorithms optimized for a specific class of images but not the unconstrained set of natural images contaminated by a variety of noise sources.

A perceived quality score is not a proxy for a perceived utility score: a decrease in perceived quality may not affect the perceived utility. In applications acquiring, processing, and/or transmitting images for use by humans, system design constraints could be relaxed to accommodate a perceived utility constraint rather than a perceived quality constraint. Consider a human that uses visual information acquired by an imaging system to maneuver a remote reconnaissance vehicle. Designing the imaging system to operate at a specified quality constraint (i.e., a minimum quality threshold) demands better resources (e.g., sensor components, communication bandwidths, memory storage, etc.) even though the perceived utility of the image may not have been compromised. Designing the system in terms of perceived utility would accommodate degradations to perceived quality without sacrificing perceived utility, demanding fewer resources and, hence, reducing costs.

Our approach to study the utility assessment task for natural images consists of two parts. In the first part, two psychophysical experiments are conducted to investigate the relationship between the quality assessment and utility assessment tasks. The first experiment collects perceived quality scores for a collection of test images, since the test images used in this study include a class of images that have not been evaluated in a formal quality assessment study. The second experiment collects perceived utility scores for the same collection of test images. The results from the experiments suggest a complex relationship between quality and utility scores dependent upon the image representation. Namely, any algorithm optimized to predict perceived quality scores cannot immediately predict perceived utility scores.

The second part of this paper evaluates the ability of several QA algorithms [START_REF] Damera-Venkata | Image quality assessment based on a degradation model[END_REF][START_REF] Wang | Multi-scale structural similarity for image quality assessment[END_REF][START_REF] Wang | Image quality assessment: From error visibility to structural similarity[END_REF][START_REF] Sheikh | Image information and visual quality[END_REF][START_REF] Chandler | VSNR: A wavelet-based visual signal-to-noise ratio for natural images[END_REF][START_REF] Carnec | Objective quality assessment of color images based on a generic perceptual reduced reference[END_REF] to predict subjective scores for either the quality or utility assessment tasks. Although the quality assessment task inspired the design of QA algorithms, the capabilities of these algorithms for the utility assessment task is unknown. Furthermore, since these algorithms seek objective scores consistent with subjective scores, a successful algorithm could model aspects of observer performance relevant to an algorithm that predicts perceived utility scores.

Among the QA algorithms evaluated, the visual information fidelity (VIF) criterion, [START_REF] Sheikh | Image information and visual quality[END_REF] which is frequently reported to provide the highest correlation with perceived quality, predicted both perceived quality scores and utility scores reasonably. The consistent performance of VIF for both the quality and utility tasks raised suspicions in light of the conclusions drawn from the psychophysical experiments. A thorough analysis of VIF revealed that it artificially emphasizes evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). A modified implementation of VIF, denoted VIF*, is described that provides statistically significant improvement over VIF for the quality assessment task and statistically worse performance than VIF for the utility assessment task. The modifications adjust the weights used to pool its objective scores across image scales.

A novel utility assessment algorithm, referred to as the natural image contour evaluation (NICE), is introduced. NICE conducts a comparison of the contours of a test image to those of a reference image across multiple image scales to score the test image. The proposed algorithm demonstrates a viable departure from traditional QA algorithms that incorporate energy-based approaches and is capable of predicting perceived utility scores. This paper has the following organization: Section 2 specifies two broad classes of image stimuli used to evaluate the relationship between quality and utility. Section 3 specifies the experimental methodology. Experiment results are presented in Section 4. Section 5 reviews QA algorithms evaluated in this paper. An analysis with regard to the capability of QA algorithms to predict subjective scores is provided in Section 6. Conclusions are presented in Section 7.

NATURAL IMAGE REPRESENTATIONS

To investigate the relationship between the quality and utility assessment tasks, test images are necessary that produce different distortions. Two broad classes of stimuli, distinguished according to the image representation model, specify the test images used. This section describes these two image representations: signal-based representations and visual-structure-preserving representations. These image representations induce distortions that are spatially correlated with the natural image and disrupt different image characteristics to deteriorate the visual information.

An image representation specifies a sequence of images corresponding to a processing algorithm and a parameter, where subsequent images in the sequence contain additional detail or information relative to the previous images. [START_REF] Rouse | Analyzing the role of visual structure in the recognition of natural image content with multi-scale SSIM[END_REF] The first image recognized by human observers on average in an image representation sequence is called the recognition threshold. The recognition threshold (RT) specifies the minimum visual information of a test image that contains sufficient visual information to be a useful surrogate for the reference image. Images whose visual information exceeds that of the RT are useful to a human observer, whereas those with less visual information than the RT have little or no use. For a natural image, a recognition threshold is defined for each image representation, and the RT for each image representation conveys the same visual information from the reference image.

The signal-based (SB) representations (cf. Figure 1(b)) correspond to a class of images encountered whose distortions are induced by quantizing wavelet subband coefficients (e.g., JPEG-2000 compression). A type of signal-based representations, denoted DCQ-J2K, is used, where quantization step-sizes are assigned according to the dynamic contrast-based quantization (DCQ) algorithm. [START_REF] Chandler | Dynamic contrast-based quantization for lossy wavelet image compression[END_REF] DCQ computes a measure of visual distortion based on the characteristics of the image, the subband, and the display to assign subband quantization stepsizes. DCQ supports visually lossless compression. A visually lossless image is visually indistinguishable from the reference. The SB representations evolve by increasing the encoding bitrate, R.

The visual-structure-preserving (VSP) representations (cf. Figures 1(c) and 1(d)) correspond to a class of images whose texture has been removed with limited disruption to object boundaries and edges. VSP representations were obtained by implementing total variation (TV) regularization via soft-thresholding of undecimated Haar wavelet coefficients in all subbands except the low-frequency residual subband. [START_REF] Rouse | Analyzing the role of visual structure in the recognition of natural image content with multi-scale SSIM[END_REF][START_REF] Rudin | Nonlinear total variation based noised removal algorithm[END_REF][START_REF] Steidl | On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs[END_REF][START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF] The VSP representations may either include or exclude low-frequency (LF) signal information. Higher-frequency signal information is believed to convey salient visual information for interpretation, so VSP representations that exclude lowfrequency signal information, denoted VSP-noLF, were generated that isolate object boundaries and edges. The VSP representations evolve by decreasing the soft thresholding parameter, τ .

METHODS: SUBJECTIVE SCORES FOR QUALITY AND UTILITY TASKS

Subjective scores for the quality assessment and utility assessment tasks were collected for test images corresponding to each image representation for five 512 × 512 grayscale natural images used in a previous study: 9 airplane, boy & cat, train, caged birds, guitarist. This section summarizes the methods used for the experiments to collect subjective responses for both assessment tasks.

Stimuli: Each image representation (i.e., DCQ-J2K, VSP-LF, and VSP-noLF) specifies a sequence of images identified by the image representation parameter value (i.e., encoding bitrate for DCQ-J2K representations and soft-thresholding parameter τ for VSP representations). A subsequence of N = 6 images is constructed to span the range of a given image representation (i.e., DCQ-J2K, VSP-LF, and VSP-noLF) for a reference image. Specifically, sequences of DCQ-J2K images were generated for each natural image corresponding to bitrates logarithmically equally spaced from R = 0.01 to R V L , where R V L denotes the bitrate of a visually lossless image as specified by DCQ. [START_REF] Chandler | Dynamic contrast-based quantization for lossy wavelet image compression[END_REF] Sequences of VSP-LF and VSP-noLF images were generated for each natural image corresponding to τ = 2048, 446, 97, 21, 5, 1.

Quality Assessment Experiment: The quality assessment experiment asks subjects to score the perceptual quality of test images from the sequences corresponding to the DCQ-J2K, VSP-LF, and VSP-noLF image representations for each reference natural image. The subjective assessment methodology for video quality (SAMVIQ) protocol provides consistent opinion scores from observers and is used for the quality assessment experiment. [START_REF]SAMVIQ -subjective assessment methodology for video quality[END_REF][START_REF] Kozamernik | Subjective quality of internet video codecs phase ii evaluations using SAMVIQ[END_REF] Observers provided opinion scores based on the perceived quality of the test images.

To alleviate observer fatigue due to prolonged evaluation sessions, the test images were partitioned into to two equally representative sets, creating two testing sessions. For each natural image, three anchor images, each from a different image representation and spanning the range of "quality", served as anchor images and appeared in both testing sessions to facilitate opinion score alignment across both sessions. Twenty-nine observers with normal or corrected-to-normal acuity participated in the experiment, and twenty-six opinion scores were collected for each test image (52 opinion scores were collected for the anchor images).

Utility Assessment Experiment: The utility assessment experiment asks subjects to select the appropriate image from a pair of images in response to the query "Which image tells you more about the content?" The images in the pairs are selected from the sequences of two different image representations (e.g., SB and VSP-noLF) † . Results from a preliminary test concluded that certain pair comparisons were unnecessary (e.g., comparing the first image of the SB sequence with the last image of the VSP-LF sequence). This reduced the number of comparisons to 54 for each natural image. Forty observers with normal or corrected-to-normal acuity participated in the experiment, and all observers provided responses for each pair of images once.

RESULTS: PERCEIVED QUALITY AND PERCEIVED UTILITY SCORES

This section first describes the generation of meaningful perceived quality scores and perceived utility scores from the raw experimental data collected in the experiments described in Section 3. A comparative analysis of the perceived quality scores and perceived utility scores concludes this section.

Generating Meaningful Perceived Quality Scores from the Raw Experimental Data

A meaningful perceived quality score, given as a mean opinion score (MOS), is computed for each test image from the opinion scores collected in the quality assessment experiment. A MOS is the arithmetic average of the individual observer opinion scores. A linear mapping between the MOSs of the anchor images from a specific testing session were mapped to the overall MOSs of these anchor images across both testing sessions. These mappings correct for deviations in observer opinion scores between sessions. These linear mappings were applied to the raw opinion scores for the individual sessions to generate overall MOSs for each test image.

Generating Meaningful Perceived Utility Scores from the Raw Experimental Data

The paired comparison test acquires subjective data to estimate p ij = P r(X i > X j ), the probability that stimulus X i "tells you more about the content" than stimulus X j . Bradley and Terry 16 specify a mathematical model that relates the p ij to a continuum of raw scale values that ranks the collection of stimuli {X i } n i=1 . We obtain raw scale values for each natural using a generalized linear model, which Critchlow and Flinger 17 demonstrate is equivalent to traditional the maximum-likelihood method used by Bradley and Terry. [START_REF] Bradley | The rank analysis of incomplete block designs I: The method of paired comparisons[END_REF] The raw scale values order test images corresponding to a single natural image but are not aligned across natural images. Meaningful utility scores across natural images were derived from the raw scale values using a linear mapping for each natural image. The linear mapping is defined such that a utility score of 0 corresponds to a natural image's recognition threshold (RT) and a utility score of 100 corresponds to any image that is visually lossless with respect to the reference natural image. The raw scale value corresponding to the RT for each image representation was linearly interpolated using the RT parameterized by the corresponding image representation parameter (i.e., R for the DCQ-J2K and τ for the VSP representations). [START_REF] Rouse | Analyzing the role of visual structure in the recognition of natural image content with multi-scale SSIM[END_REF] The RTs were obtained in a previous study. [START_REF] Rouse | Analyzing the role of visual structure in the recognition of natural image content with multi-scale SSIM[END_REF] The raw scale values corresponding to the RTs were consistent across the image representations; therefore, the interpolated raw scale values corresponding to the RT for each image representation were averaged to obtain the raw scale value corresponding to the RT of the natural image. Using the raw scale value corresponding to the RT of the natural image and the raw scale value corresponding to the DCQ-J2K visually lossless image, a linear mapping from raw scale values to meaningful utility scores is obtained. Test images with perceived utility scores below -10 were omitted from set of test images, leaving 62 test images.

Comparative Analysis Between Perceived Quality and Perceived Utility

A perceived quality score is not a proxy for a perceived utility score. Figure 2 shows the relationship between the perceived utility scores as a function of the perceived quality scores plotted by natural image and by image representation. Although the relationship appears linear, use of a linear fit to map perceived quality scores to perceived utility scores will produce poor utility estimates (RMSE = 15.1). Perceived utility exhibits an interesting relationship with perceived quality. Namely, the linear relationship between quality and utility for images with quality scores below 30 suggests that observers judge very low quality images in terms of the ability to interpret the content. Furthermore, for images with perceived quality scores above 40, the VSP-noLF images have nearly equal perceived utility scores to their VSP-LF counterparts (i.e., equal values of τ ), yet many of the VSP-noLF images have significantly lower perceived quality scores (about 25 quality points lower) than their VSP-LF counterparts. In general, any algorithm optimized to predict subjective quality scores cannot immediately predict subjective utility scores.

FULL-REFERENCE QUALITY ASSESSMENT ALGORITHMS

This section reviews the full-reference QA algorithms investigated for their ability to predict the subjective quality scores and subjective utility scores. Full-reference quality assessment algorithms harness the availability of an explicit, external reference image X to predict the subjective score of a test image X, and serve as a tool to compare the two images. These full-reference QA algorithms can be categorized as 1) conventional distortion measures, 2) algorithms based on properties of the HVS, and 3) algorithms derived from hypothetical high-level HVS objectives. While many QA algorithms based on hypothetical high-level HVS objectives produced objective scores consistent with perceived quality scores, this work specifically focuses on the visual information fidelity (VIF) criterion and a novel algorithm refer to as the natural image contour evaluation (NICE). A mathematical description of these two QA algorithms is included this section.

Conventional Signal Fidelity Measures

Mean-square error (MSE), which is used to compute peak signal-to-noise ratio (PSNR), and root-mean squared (RMS) distortion contrast provide computationally simple evaluations of signal fidelity. These measures evaluate fidelity solely in terms of the energy of the distortions. Root-mean-squared (RMS) distortion contrast C rms (E) evaluates fidelity based on the visibility of the distortions E = X -X when comparing the images on a particular display device. [START_REF] Poynton | The rehabilitation of gamma[END_REF]

Algorithms Based on Properties of the Human Visual System

Several quality assessment algorithms capitalize on models and principles characterizing low-level HVS properties such as contrast sensitivity, [START_REF] Valois | Spatial Vision[END_REF] contrast masking, [START_REF] Valois | Spatial Vision[END_REF][START_REF] Legge | Contrast masking in human vision[END_REF][START_REF] Chandler | Effects of natural images on the detectability of simple and compound wavelet subband quantization distortions[END_REF] and perceived contrast. [START_REF] Georgeson | Contrast constancy: Debluring in human vision by spatial frequency channels[END_REF][START_REF] Brady | What's constant in contrast constancy? The effects of scaling on the perceived contrast of bandpass patterns[END_REF] These properties model the detection of a visual target (e.g., the distortions in an image) under a variety of conditions based on the contrast of the distortions. Many quality assessment algorithms have been proposed, [START_REF] Stockham | Image processing in the context of a visual model[END_REF][START_REF] Mannos | The effects of a visual fidelity criterion on the encoding of images[END_REF][START_REF] Pearlman | A visual system model and a new distortion measure in the context of image processing[END_REF][START_REF] Granrath | The role of human visual models in image processing[END_REF][START_REF] Safranek | A perceptually tuned sub-band image coder with image dependent quantization and post-quantization data compression[END_REF][START_REF] Daly | The visible difference predictor: An algorithm for the assessment of image fidelity[END_REF][START_REF] Lubin | The use of psychophysical data and models in the analysis of display system performance[END_REF][START_REF] Watson | DCT quantization matrices visually optimized for individual images[END_REF][START_REF] Teo | Perceptual image distortion[END_REF][START_REF] Watson | Visibility of wavelet quantization noise[END_REF][START_REF] Pappas | Perceptual criteria for image quality evaluation[END_REF] but this section summaries a subset that represents a variety of approaches.

The weighted SNR (WSNR) and noise quality measure (NQM) algorithms evaluate images by incorporating HVS properties to first simulate the appearance of the reference and test images to a human, and then, the SNR is computed based on the difference of the simulated images. [START_REF] Damera-Venkata | Image quality assessment based on a degradation model[END_REF] The visual signal-to-noise ratio (VSNR) algorithm evaluates images according to a contrast model accounting for low-level HVS properties and the mid-level HVS property of global precedence. [START_REF] Chandler | VSNR: A wavelet-based visual signal-to-noise ratio for natural images[END_REF][START_REF] Navon | Forest before trees: The precedence of global features in visual perception[END_REF] The criterion 4 (C4) algorithm assesses images using elaborate models of several processing areas of the visual cortex. [START_REF] Carnec | Objective quality assessment of color images based on a generic perceptual reduced reference[END_REF] The models in C4 describe color vision; frequency-orientation analysis; contour detection; perceptual and localization of patterns; object discrimination; and visual memory.

Algorithms based on Hypothetical Objectives of the Human Visual System

A family of QA algorithms has been developed based on the premise that the HVS has evolved in response to the statistical regularities exhibited by the physical world, and, thus, produces objective scores according to the resemblance of a test image's structural information (i.e., edges and object boundaries) to that of the reference image. [START_REF] Sheikh | An information fidelity criterion for image quality assessment using natural scene statistics[END_REF] Algorithms from this family include the structural similarity (SSIM) metric, 5 the information fidelity criterion (IFC), [START_REF] Sheikh | An information fidelity criterion for image quality assessment using natural scene statistics[END_REF] and the visual information fidelity (VIF) criterion. [START_REF] Sheikh | Image information and visual quality[END_REF] This section first summarizes these QA algorithms. Second, this section presents the mathematical specification of VIF. This section concludes with a discussion and specification of a modified version of VIF, denoted VIF*.

Summary of SSIM, MS-SSIM, and VIF

The structural similarity (SSIM) 5 metric employs a local measure of spatial correlation between the pixels of the reference and test images that is modulated by distortions quantified by locally normalized first (mean) and second (variance) moments. MS-SSIM extends SSIM by evaluating this modified spatial correlation measure across several image scales. [START_REF] Wang | Multi-scale structural similarity for image quality assessment[END_REF] Two of the authors present extended discussions of SSIM and MS-SSIM elsewhere. [START_REF] Rouse | Analyzing the role of visual structure in the recognition of natural image content with multi-scale SSIM[END_REF][START_REF] Rouse | Understanding and simplifying the structural similarity metric[END_REF] The information fidelity criterion (IFC) [START_REF] Sheikh | An information fidelity criterion for image quality assessment using natural scene statistics[END_REF] and the visual information fidelity (VIF) criterion 6 generate objective scores based on a measurement of the mutual information between the test and reference image. Both use fundamentally Gaussian models of the wavelet coefficients of the test image and reference image which reduces the mutual information measurement to a local signal-to-noise ratio in the wavelet domain (cf. Eq. ( 2)).

Mathematical Specification of VIF

This section reviews VIF, which is an extension of IFC that incorporates a simple HVS model. [START_REF] Sheikh | Image information and visual quality[END_REF][START_REF] Sheikh | An information fidelity criterion for image quality assessment using natural scene statistics[END_REF] The HVS is modeled as an additive Gaussian noise source attributed to low-level HVS processing. [START_REF] Sheikh | Image information and visual quality[END_REF] The assessment of a test image is based on spatially local SNR measurements, computed at multiple image scales, of both the reference and test images contaminated with the modeled, low-level HVS noise. Let the elements of the length N k vectors C k and D k denote the wavelet coefficients of the k th subband of the reference and test images, respectively ‡ . The elements of the length N k vectors E k and F k denote the wavelet coefficients of the k th subband of the respective reference and test images that have been contaminated with visual noise.

VIF parses each wavelet subband into disjoint blocks composed of P coefficients. The following discussion assumes only one subband, so the superscript k is omitted in the subsequent discussion. Let C b and D b correspond to the b th block of P spatially adjacent coefficients of C and D, respectively. The b th block of wavelet coefficients in the subband of the reference image may be modeled as a Gaussian scale mixture [START_REF] Wainwright | Scale mixtures of gaussians and the statistics of natural images[END_REF][START_REF] Wainwright | Random cascades on wavelet trees and their use in analyzing and modeling natural images[END_REF] (GSM) random vector given as C b = s b U, where s b is a positive random scalar and U is a zero mean Gaussian random vector of length P with covariance K U . Given s b , the coefficient block C b is a zero mean Gaussian random scalar with covariance I + σ 2 N I VIF combines two evaluations to yield an overall assessment of a test image. First, an evaluation comparing the reference coefficients before and after the HVS model value is computed. Second, an evaluation comparing the reference coefficients before the HVS model to the processed coefficients after the HVS model is computed. These two evaluations are computed for each wavelet subband. The ratio of the sum of these evaluations across the subbands provides an overall assessment of the test image. Let s be a length B k vector whose b th element is s b . Given s, the VIF value is given by

VIF = K k=1 IF C(C k , F k ) K k=1 IF C(C k , E k )
.

(

) 1
The terms IF C(C k , F k ) and IF C(C k , E k ) are based on IFC [START_REF] Sheikh | An information fidelity criterion for image quality assessment using natural scene statistics[END_REF] and are defined as

IF C(C k , F k ) = B k b=1 log 2 |g 2 b s 2 b K U + (σ 2 V b + σ 2 N )I| |(σ 2 V b + σ 2 N )I| IF C(C k , E k ) = B k b=1 log 2 |s 2 b K U + σ 2 N I| |σ 2 N I| , (2)
where | • | denotes the matrix determinant and the terms g b , s b , K U , and σ V b vary with k and are computed from

C k and D k . For subband k, the term g b is estimated as the linear regression of block D b on the block C b , and the variance of the additive zero mean Gaussian noise V b is the mean squared error of the regression.

VIF* Specification

VIF artificially emphasizes evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). Thus, VIF is invariant to changes to low spatial frequency energy, which distinguishes the VSP-LF and VSP-noLF representations. ) corresponding to finer scales are an order of magnitude larger than those at coarser scales. Thus, VIF cannot produce distinct scores that reflect the changes in the perceived quality scores for these images.

The problem identified with VIF (cf. Figure 3) is due to the variation in the number of coefficients blocks B k for subbands at different image scales. Specifically, subbands corresponding to finer image scales have more § All of these test images correspond to the same reference image, so the denominator IF C(C k ; E k ) is irrelevant to a discussion of VIF and merely serves as a scale factor unique to each reference image. wavelet coefficients that subbands corresponding to coarser image scales due to the use of a decimated wavelet transform; for a fixed block size P , the number of coefficient blocks is smaller for subbands corresponding to coarser image scales. A modified definition of VIF, denoted VIF*, is given that scales the calculation for each subband by the number of blocks B k for that subband. VIF* is given as

VIF * = K k=1 1 B k IF C(C k , F k ) K k=1 1 B k IF C(C k ; E k ) , (3)
where IF C(C k ; F k ) and IF C(C k ; E k ) are defined as in Eq. ( 2). Figure 3(b) shows the value of 1 B k IF C(C k ; F k ) (i.e., the numerator of VIF*) for the subbands capturing horizontal edges for different image scales for the same three images corresponding to the airplane image described in Figure 3(a). VIF* reveals greater differences between the images of the VSP-LF and VSP-noLF representations corresponding to the coarsest scale (scale 4). Thus, VIF* produces distinct scores that reflect the changes in the perceived quality scores for these images.

NICE: Natural Image Contour Evaluation

This section describes the proposed natural image contour evaluation (NICE) algorithm for utility assessment. NICE compares the contours of a test image to that of a reference image across multiple image scales to produce a numerical score of the test image. This novel algorithm demonstrates a viable departure from traditional QA algorithms that incorporate energy-based approaches.

A wavelet representation of an image provides multiscale directional derivatives of that image, which can be used to identify image contours at different image scales. For NICE, both the reference and test image are represented using an undecimated implementation of the steerable pyramid 40 using D orientations and S scales ¶ . Let W s,θ (i) and Ŵs,θ (i) denote the i th wavelet coefficient of the respective reference and test images in the subband corresponding to scale s ∈ {1, 2, . . . , S} and orientation θ ∈ {0, π D , 2π D , . . . , π(D-1) D }. The remainder of this section describes the algorithm to identify image contours and the comparison of those contours to generate a NICE score for the test image.

Image Contour Identification from Local Modulus Maxima

For each image scale s, the local modulus maxima [START_REF] Mallat | Characterization of signals from multiscale edges[END_REF] of wavelet coefficient scales are used to identify image contours for the reference and test images. The local modulus maxima are determined from gradient vectors formed from wavelet subbands corresponding to derivatives in horizontal and vertical spatial directions. [START_REF] Mallat | Characterization of signals from multiscale edges[END_REF] Define G s (i) = W s,0 (i) -jW s, π 2 (i) and Ĝs (i) = Ŵs,0 (i)j Ŵs, π 2 (i) as the gradient of the respective reference and test images at scale s, where j = √ -1. For image scale s, let M s (i) = |G s (i)| and A s (i) = G s (i) denote the respective modulus and angle of the gradient of the reference image. Similarly, define Ms (i) = | Ĝs (i)| and Âs (i) = Ĝs (i) for the test image. Local modulus maxima of the reference image correspond to points of M s (i) greater than the two adjacent neighbors in the direction indicated by A s (i), and for the test image, local modulus maxima are similarly identified using Ms (i) and Âs (i). Let I s denote the set of indices i corresponding to local modulus maxima of the reference image at scale s, and let Îs denote the set of indices i corresponding to local modulus maxima of the test image at scale s.

Binary images represent image contours of the reference and test images. The image contours at scale s of the reference and test images are identified as local modulus maxima that exceed a threshold β s based on the reference image and is given as

β s = 1 p max i M s (i), (4)
for some scalar p > 0. Specifically, define B s (i) and Bs (i) as

B s (i) = 1 M s (i) > β s and i ∈ I s 0 else Bs (i) = 1 Ms (i) > β s and i ∈ Îs 0 else . (5)

NICE Score

An objective score based for NICE is computed by comparing the contours of the reference and test images at each image scale s, represented as the respective binary images B s (i) and Bs (i) defined in Section 5.4.1.

The binary images B s (i) and Bs (i) are subjected to morphological dilation 42 with a 3 × 3 "plus-sign" shaped structuring element, and the point-wise exclusive-or operation of the dilated binary images produces the binary image E s (i). The overall NICE score for the test image is computed as

NICE = S s=1 a s N (s), (6)
where N (s) denotes the number of non-zero elements of E s and the {a s } S s=1 are nonnegative scalars.

ANALYSIS AND DISCUSSION

Although the psychophysical evidence presented in Section 4 establishes a complex relationship between scores for quality task and utility task, quality assessment (QA) algorithms, which are typically developed for the quality task, could be useful to predict subjective utility scores. The QA algorithms and the proposed utility assessment algorithm described in Section 5 are evaluated in terms of the their capability to predict subjective scores of test images for both the quality and utility assessment tasks. VIF, VIF*, and NICE use a four-scale steerable pyramid with six orientations, but only subbands corresponding to horizontal and vertical frequencies are used. The implementations for most of the algorithms were obtained from the respective authors of the algorithms.

Objective scores generated by QA algorithms frequently exhibit a nonlinear relationship with subjective scores. Objective scores are fitted to subjective scores with a third order polynomial subject to a monotonicity constraint. The fitted objective scores are evaluated with respect to the subjective scores using the Spearman rank order correlation coefficient (ROCC), the squared Pearson (linear) correlation coefficient r 2 , the root mean squared error (RMSE), an F -test to individually compare the residual variance of NICE to the other algorithms, and the outlier ratio (OR). The OR is the proportion of fitted objectives scores that differ from the subjective scores by more than twice the subjective score's estimated standard error.

An F -test determines whether the residual variance of algorithm is statistically larger or smaller than the other. [START_REF] Devore | Probability and Statistics for Engineering and the Sciences[END_REF] Suppose σ 2

A and σ 2 B denote the variance of the residuals corresponding to algorithms A and B when Table 1. Results summarizing the performance of quality assessment algorithms and proposed NICE utility assessment for the quality assessment and utility assessment tasks. NICE is designed for the utility assessment task and not the quality assessment task. Results on its performance for the quality assessment task demonstrate the divergence between performance for the quality and utility assessment tasks when utility is predicted well (cf. Section 4.3). Objective scores, fitted to subjective scores with a monotonic third order polynomial, are evaluated with respect to the subjective scores using the Spearman rank order correlation coefficient (ROCC), the squared Pearson (linear) correlation coefficient r 2 , the root mean squared error (RMSE), an F -test to individually compare the residual variance of NICE to the other algorithms, and the outlier ratio (OR). Values of Fstatistic > F critical = 1.53 (or Fstatistic < 1/F critical = 0.654) have statistically larger (or smaller) residual variances than the proposed algorithm at the 95% confidence level. , the F -test computes a critical value F critical based on the number of residuals and the confidence level (1 -α)% of the test, and values of F statistic > F critical (or F statistic < 1/F critical ) have statistically larger (or smaller) residual variances than the proposed algorithm at the (1 -α)% confidence level. For a 95% confidence level and 62 test images, F critical = 1.53. This section first evaluates the ability of the QA algorithms and the proposed NICE utility assessment algorithm to the predict subjective scores for the quality and utility assessment tasks. NICE is designed for the utility assessment task and not the quality assessment task. Results on its performance for the quality assessment task demonstrate the divergence between performance for the quality and utility assessment tasks when utility is predicted well (cf. Section 4.3). Second, this section investigates the benefits of appropriate pooling across image scale evaluations with VIF and NICE to predict subjective scores for both assessment tasks.

Quality Task

Predicting Perceived Quality Scores and Perceived Utility Scores with Quality Assessment Algorithms

Among the QA algorithms evaluated, VIF* provides significantly smaller errors (RMSE = 4.22) for predicting perceived quality scores than the other QA algorithms. Table 1 summarizes the results from a statistical analysis of the fitted QA algorithm scores with the perceived quality scores This significance of the improvement with VIF* was established using an F -test to compare the residual variances of the fitted VIF* scores to the fitted scores of other algorithms. The performance improvement with VIF* for the quality task, validates the proposed modifications to VIF.

Overall, the QA algorithms based on hypothetical properties of the HVS and the proposed NICE utility assessment algorithm provide better predict perceived quality scores than the other classes of algorithms. An inspection of the QA algorithms classified as either conventional signal fidelity measures or based on properties of the HVS revealed that these algorithms assigned relatively low scores to images corresponding to the VSP-noLF representation, which suggests that they overemphasize the relevance of distortions to low spatial frequencies.

For the utility assessment task, both VIF and NICE provide significantly smaller errors (RMSE = 7.32 and 8.94, respectively) for predicting perceived utility scores than the other algorithms. Table 1 summarizes the statistical analysis of the fitted algorithm scores with the perceived utility scores. The variance of the residuals corresponding to fitted objective scores produced by VIF and NICE were found to be statistically equivalent

Improving Pooling Strategies Across Image Scales to Predict Subjective Scores

The HVS has been characterized as a multi-scale analyzer of visual data. [START_REF] Valois | Spatial Vision[END_REF] Thus, many QA algorithms evaluate a test image with respect to a reference image at multiple image scales. In light of the improvement of VIF* over VIF for the quality task, this section demonstrates the benefits of properly pooling multi-scale evaluations of VIF and NICE using the optimal linear estimator of the subjective scores given the objective scores for each image scale. This approach is compared to using only single image scale objective scores.

Given objective scores computed for each image scale of either VIF † or NICE, the optimal linear estimator of the subjective scores is found after fitting the single scale objective scores to the subjective scores with a third order polynomial subject to a monotonicity constraint. The polynomial fit resolves the nonlinear relationship between the objective scores and the subjective scores.

The optimal pooling weights improve the predictive capabilities of VIF for both the quality task (RMSE = 4.01) and the utility task (RMSE = 7.13); the optimal pooling weights improve NICE for the the utility task (RMSE = 7.04) and exhibit essentially the same performance for the quality task. For both algorithms, the optimal weights differ for each assessment task. Table 2 summarizes the performance analysis of VIF and NICE using individual image scale objective scores and the minimum mean squared error (MMSE) linear estimator that combines objective scores from each image scale. Table 3 lists the optimal weights for VIF and NICE for the quality and utility task. The weights have been normalized such that the absolute maximum weight for each algorithm and task is one.

With only 62 test images available, the number of parameters used to fit the objective scores to the subjective data restrict the inferences one can make from these results. Each nonlinear fit requires four parameters for each scale, and the addition of the four weights yields to a total of 20 parameters. Despite this shortcoming, these results advocate the potential benefits of combining objective scores across image scales.

CONCLUSIONS AND FUTURE WORK

This paper investigates the relationship between the quality assessment and utility assessment tasks for natural images. Two psychophysical experiments are conducted to collect perceived quality scores and perceived utility scores for a collection of test images corresponding to signal-based representations and visual-structure-preserving representations. The results from these experiments provide evidence that any quality assessment (QA) algorithm optimized to predict perceived quality scores cannot immediately predict perceived utility scores.

Several QA algorithms are evaluated in terms of their ability to predict subjective scores for the quality and utility assessment tasks. An analysis of VIF, instigated by its suspiciously consistent performance for both tasks, revealed that it artificially emphasized evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). A proposed implementation of VIF, denoted VIF*, demonstrates statistically significant improvement over VIF for the quality assessment task and statistically worse performance for the utility assessment task. NICE, a novel utility assessment algorithm is introduced that conducts a comparison of the contours of a test image to those of a reference image across multiple image scales to score the test image. NICE demonstrates a viable departure from traditional QA algorithms that incorporate energy-based approaches and is capable of predicting perceived utility scores. This investigation established several encouraging areas of future work. First, while formulating a relationship between perceived quality scores and perceived utility scores based on image representations is useful, it is hypothesized that a more appropriate relationship between could be modeled as the perceptual change to an image when removing lower spatial frequency content. Second, though NICE establishes a novel framework that provides reasonable prediction of perceived utility scores, a rigorous evaluation of its parameters and components could yield significant improvements. Last, the demonstrated benefits of linear pooling to combine objective scores across image scales urge an extensive analysis of other pooling strategies.

Figure 1 .

 1 Figure 1. Original reference airplane and average observer recognition thresholds for two types of visual-structurepreserving (VSP) representations and the DCQ signal-based representation (DCQ-J2K). The VSP representations shown differ with regard to the inclusion of low-frequency (LF) information contained in the low-frequency residual subband.

  Subjective scores plotted by image representation.

Figure 2 .

 2 Figure 2. Scatter plots showing the relationship between perceived utility scores and the perceived quality scores for five natural images. Standard error bars have been included for the subjective scores.

s 2 b

 2 K U , and C b is conditionally independent of C m for all m = b. VIF relates the b th block of wavelet coefficient of the test and reference images according to the linear model D b = g b C b + V b , where g b is a deterministic scalar defined for each block and V b is a zero mean Gaussian random vector of length P with covariance matrix σ 2 V b I specified for each block b. Thus, given s b , the block of coefficients D b is also a Gaussian random vector with covariance g 2 b s 2 b K U + σ V b I. Independent zero-mean additive Gaussian noise sources model low-level HVS noise in VIF; coefficients of the reference and test images are contaminated with visual noise. Let E b and F b correspond to the b th block of P spatially adjacent coefficients of E and F , respectively. The output of the HVS model for the reference image is E b = C b + M b , and the output of the HVS model for the test image is F b = D b + N b . The terms M b and N b are a zero mean Gaussian random vectors of length P with covariance σ 2 M I = σ 2 N I, where σ 2 N = σ 2 M is the HVS model parameter. Thus, given s b , the block of coefficients E b is a Gaussian random vector with covariance s 2 b K U + σ 2 N I, and the block of coefficients F b is also a Gaussian random vector with covariance g 2 b s 2 b K U + σ 2 V b

Figure 3 (

 3 a) demonstrates this characteristic of VIF for the airplane image under the DCQ-J2K (MOS = 65.9), VSP-LF (MOS = 68.6), and VSP-noLF (MOS = 53.5) image representations. In particular, Figure 3(a) shows the values of IF C(C k , F k ) (i.e., the numerator § of VIF) for the subband capturing horizontal edges as a function of image scale k. The values of IF C(C k , F k

Figure 3 .

 3 Figure 3. VIF artificially emphasizes evaluations at finer image scales (i.e., higher spatial frequencies) over those at coarser image scales (i.e., lower spatial frequencies). Thus, VIF is invariant to changes to low spatial frequency energy, which distinguishes the VSP-LF and VSP-noLF representations. Figure3(a) demonstrates this characteristic of VIF for the airplane image under the DCQ-J2K (MOS = 65.9), VSP-LF (MOS = 68.6), and VSP-noLF (MOS = 53.5) image representations. The value of the numerator of VIF for the subband capturing horizontal edges is shown at different image scales. VIF* resolves this problem with VIF and is demonstrated in Figure 3(b) for the same three images.

Table 2 .

 2 A summary of the prediction abilities of VIF and the proposed NICE utility assessment algorithm for the quality and utility assessment tasks using individual image scale objective scores and the minimum mean squared error (MMSE) linear estimator that combines objective scores from each image scale. Refer to Table1for a description of the column acronyms.

			Quality Task			Utility Task
	Algorithm	r 2	ROCC RMSE	OR	r 2	ROCC RMSE	OR
	VIF (scale 1)	0.861 0.911	8.61	0.597 0.959	0.966	7.35	0.371
	VIF (scale 2)	0.895 0.950	7.48	0.532 0.959	0.961	7.32	0.387
	VIF (scale 3)	0.967 0.978	4.22	0.339 0.917	0.920	10.5	0.532
	VIF (scale 4)	0.802 0.876	10.3	0.613 0.602	0.718	22.9	0.710
	VIF (MMSE)	0.971 0.979	4.01	0.323 0.962	0.970	7.13	0.371
	NICE (scale 1)	0.788 0.882	10.6	0.855 0.844	0.930	14.3	0.581
	NICE (scale 2)	0.893 0.943	7.56	0.581 0.954	0.958	7.82	0.419
	NICE (scale 3)	0.854 0.922	8.81	0.597 0.805	0.847	16.0	0.629
	NICE (scale 4)	0.676 0.773	13.2	0.790 0.476	0.608	26.9	0.790
	NICE (MMSE) 0.923 0.958	6.67	0.581 0.963	0.966	7.04	0.387

Table 3 .

 3 Weights to linearly combine objective score from each image scale found by minimizing the mean squared error between the combined objective scores and subjective scores for each task.

			Quality Task			Utility Task
	Algorithm	scale 1 scale 2 scale 3 scale 4 scale 1 scale 2 scale 3 scale 4
	VIF (MMSE)	0.46	-0.39	1	0.10	0.74	1	-0.18	0.06
	NICE (MMSE)	0.38	1	0.68	0.26	0.02	1	-0.25	0.02

based on an F -test. The inability of VIF*, as well as the other QA algorithms, to produce accurate objective utility scores indicates that an algorithm capable of predicting perceived utility scores should not overemphasize degradations to energy at lower spatial frequencies.

Appears in SPIE: Human Vision and Electronic Imaging 2009

† It is assumed that the ordering of the image representation sequences also corresponds to the utility of the images within an image representation.Appears in SPIE: Human Vision and Electronic Imaging 2009

‡ The subscript k for N k accounts for decimated wavelet decompositions, such as the steerable pyramid, whose subbands in coarser image scales have fewer coefficients than subbands in finer image scales.Appears in SPIE: Human Vision and Electronic Imaging 2009

¶ The high-pass residual generated by the steerable pyramid is not used to compute the NICE score.Appears in SPIE: Human Vision and Electronic Imaging 2009

† The modifications of VIF that specify VIF* do not change the results using single image scale objective scores.Appears in SPIE: Human Vision and Electronic Imaging 2009

ACKNOWLEDGEMENTS

The authors would like to thank John C. Handley of Xerox for his gracious assistance with deriving scale values from our paired comparison experiment data.


Yann Rébillé
email: yann.rebille@univ-nantes.fr

Autocontinuity and convergence theorems for the Choquet integral

Keywords: Choquet integral, null-additive set functions, autocontinuity

Our aim is to provide some convergence theorems for the Choquet integral with respect to various notions of convergence. For instance, the dominated convergence theorem for almost uniform convergence is related to autocontinuous set functions. Autocontinuity can also be related to convergence in measure, strict convergence or mean convergence. Whereas the monotone convergence theorem for almost uniform convergence is related to monotone autocontinuity, a weaker version than autocontinuity.

Introduction

Convergence theorem for sequences of measurable functions play a central role in classical measure theory. Convergence in mean or in measure for instance are traditionally related to the sigma-additivity of a measure, a continuity condition. Generalizations of classical measure theory can be pursued in the direction of non-additive set functions such as fuzzy measures. A key property of additivity is null additivity (Pap [START_REF] Pap | Null-additive set functions[END_REF]), whereas sigma-additivity can be associated to strong continuity. An intermediate notion introduced by Wang [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF] is the one of autocontinuity. Autocontinuity guarantees null-additivity and contains some regularity conditions related to continuity but does not necessarily entail strong continuity. Wang's article explores the validity of classical convergence theorem for sequences of measurable functions in term of their fuzzy integrals. A similar approach is pursued with respect to non-additive integration in term of Choquet integrals, a generalization of the standard Lebesgue integral ( [START_REF] Wang | Convergence theorems for sequences of Choquet integrals[END_REF]). Classical theorems such as the monotone or the dominated convergence theorem are established therein. Our aim is to identify which properties a set function might possess when we consider different types of convergence: -convergence in measure (µ), almost everywhere convergence (a.e.), almost uniform (a.u.) convergence or convergence in mean (m), and different modes of convergence: -monotone, dominated. As it turns out, dominated convergence theorems with a.u. convergence are unsurprisingly related to autocontinuity. Still, autocontinuity can also be characterized through µ convergence, m convergence and in a more direct way through strict convergence. Whereas dominated convergence theorems with a.e. convergence are related to strong continuity. As for the monotone convergence theorems with a.u. convergence, they are related to a weaker version of autocontinuity, namely: autocontinuity from below or above. These properties trace back to [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF]. Next section introduces the relevant material for our study. In section 3 we present various types of convergence theorems whether we consider a.u., a.e. or µ convergence. Section 4 restates our results by duality. Finally, we illustrate through examples that autocontinuity, order continuity and monotone autocontinuity are independent properties.

Preliminaries

Let (Ω, A) be a measurable space and µ a set function, µ : A → IR. From now on we will assume that µ(∅) = 0. µ is monotone if for all A, B ∈ A, µ(A) ≤ µ(B) whenever A ⊂ B. We define the conjugate set function µ by µ(A) = µ(Ω)µ(A c ). A sequence {f n } n of finite-valued measurable functions converges almost everywhere to f if there is a set E with µ(E) = 0 such that the sequence {f n 1l E c } n converges to f 1l E c . The sequence {f n } n converges almost uniformly to f if for all ǫ > 0 there is a set E ǫ with µ(E ǫ ) < ǫ such that the sequence {f n 1l E c ǫ } n converges uniformly to f 1l E c ǫ . We shall write as usual f n a.e.

-→ f and f n a.u.

-→ f respectively, and ↓, ↑ if the convergence is monotonically non-increasing, non-decreasing.

Let f be a measurable bounded function and µ a monotone set function. The Choquet integral of f ( [START_REF] Choquet | Théorie des capacités[END_REF]) with respect to µ is given by

f dµ = ∞ 0 µ({f ≥ t}) dt + 0 -∞ µ({f ≥ t}) -µ(Ω) dt
In the integrand, the large inequalities may be replaced by strict ones since µ is monotone. The computation with respect to the conjugate set function is given by f dµ = --f dµ. We will focus our study to bounded measurable functions, a set large enough to deal with usual applications where the existence of the Choquet integral is always guaranteed. We denote by B ∞ (Ω) (B ∞ (Ω) + ) the set of (non-negative) bounded and measurable functions defined on Ω.

Two functions f, g are said to be comonotonic or compatible if for all ω, ω ′ ∈ Ω, f (ω) > f (ω ′ ) ⇒ g(ω) ≥ g(ω ′ ) ( [START_REF] Denneberg | Non-additive measure and integral[END_REF][START_REF] Murofushi | An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure[END_REF][START_REF] Schmeidler | Integral representation without additivity[END_REF]). In that case, f + g dµ = f dµ + g dµ.

Let us recall three properties that link continuity condition of a set function to the convergence of a sequence of functions and their Choquet integrals.

Property 2.1 Let µ be a set function. Then, µ is monotone if and only if for all

{f n } n ∪ {f } ⊂ B ∞ (Ω) if f n ↓ u. (↑ u. )f then 1 f n dµ ↓ (↑) f dµ.
In order to extend Property 2.1 to everywhere convergence some further continuity conditions on the set function are introduced.

µ is said to be order continuous if µ(A n ) ↓ 0 whenever A n ↓ ∅. µ is be continuous from above if for all A n ↓ A then µ(A n ) ↓ µ(A)
. By definition a continuous from above set function is monotone (take A 1 = A, A n = B for n ≥ 2) and order continuous. µ is said to be continuous from below if for all

A n ↑ A then µ(A n ) ↑ µ(A).
Property 2.2 Let µ be a set function. Then, µ is monotone and order continuous if and only if for all

{f n } n ⊂ B ∞ (Ω), α ∈ IR if f n ↓ e. α.1l Ω then f n dµ ↓ αµ(Ω).
Property 2.3 Let µ be a monotone set function. Then, µ is monotone continuous from above (below) if and only if for all

{f n } n ∪ {f } ⊂ B ∞ (Ω) if f n ↓ e.
(↑ e. )f then f n dµ ↓ (↑) f dµ.

Our aim is to extend Properties 2.1 and 2.3 to a.u. and a.e. convergence. To deal with these types of convergence we must recall some facts about nulladditivity.

A set E is said to be null if for all F, µ(E ∪ F ) = µ(F ). µ is said to be null-additive if µ(E ∪ F ) = µ(F ) for all F whenever µ(E) = 0, or equivalently µ(F ) = µ(F \ E) for all F whenever µ(E) = 0. This way a set function µ is null-additive if and only if for all E, E is null as soon as µ(E) = 0. The following proposition characterizes null-additivity in terms of Choquet integrals ( [START_REF] Pap | Null-additive set functions[END_REF][START_REF] Murofushi | A Theory of Fuzzy Measures: Representations, the Choquet Integral, and Null Sets[END_REF]), Proposition 2.1 Let µ be a monotone set function. Then,

µ is null-additive ⇐⇒ if µ(E) = 0 then for all f ∈ B ∞ (Ω) + , f dµ = f 1l E c dµ ⇐⇒ if µ(E) = 0 then for all f, g ∈ B ∞ (Ω) + , f 1l E c = g1l E c ⇒ f dµ = gdµ.
1 (if). We can observe that the Choquet integral is always defined for indicator functions, 1l C dµ = µ(C) for all C ∈ A. Thus A ⊂ B ⇐⇒ 1l A ≤ 1l B ⇒ µ(A) ≤ µ(B), so µ is monotone.

Proof: For the first equivalence, (⇐) is straightforward with

f = 1l F + 1l E with E ∩ F = ∅. (⇒) Let t > 0. Then, {f ≥ t} = {f ≥ t} ∩ E + {f ≥ t} ∩ E c = {f 1l E ≥ t} + {f 1l E c ≥ t}.
Now since µ(E) = 0 we have also have µ({f 1l E ≥ t}) = 0 for all t > 0. And this gives by null-additivity,

µ({f 1l E c ≥ t}) = µ({f ≥ t})
and we conclude by integrating on [0, ∞), f dµ = f 1l E c dµ. For the second equivalence, (⇐) is straightforward with g = f 1l E c . For (⇒) we have,

f dµ = f 1l E c dµ = g1l E c dµ = gdµ. ⊓ ⊔
The relationship between order continuity, null-additivity is thoroughly studied in [START_REF] Asahina | Relationship among continuity conditions and null-additivity conditions in non-additive measure theory[END_REF] among other properties such as strong order continuity, exhaustivity or null-continuity.

3 Convergence theorems

Almost everywhere convergence

We will now focus on Property 2.3 and extend it to monotone a.e. convergence.

µ is said to be strongly continuous from above if for all E, F n such that µ(E) = 0 and

F n ↓ F with F n ∩ E = ∅ then µ(E ∪ F n ) ↓ µ(F ).
In particular µ is strongly order continuous i.e., for all A n ↓ A with µ(A) = 0 then µ(A n ) ↓ 0 (let F n = A n \ A and E = A). By definition a strongly continuous from above set function is monotone (take E = ∅, F 1 = A, F n = B for n ≥ 2). Proposition 3.1 Let µ be a set function. Then, µ is strongly continuous from above ⇐⇒ µ is continuous from above and null-additive.

Proof: (⇒) From the definition of strong continuity from above with E = ∅ we obtain continuity from above. With F n = F we obtain null-additivity. (⇐) Let E, F n be such that µ(E) = 0 and

F n ↓ F with F n ∩ E = ∅. We have µ(E ∪ F n ) ↓ µ(E ∪ F ) by continuity and µ(E ∪ F ) = µ(F ) by null-additivity. ⊓ ⊔ Theorem 3.1 Let µ be a set function. Then, µ is strongly continuous from above ⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n ↓ a.e. f ⇒ f n dµ ↓ f dµ Proof: Monotonicity is immediate. (⇐) Let µ(E) = 0 and F n ↓ F with F n ∩ E = ∅. Then 1l E∪Fn ↓ a.e. 1l F , thus µ(E ∪ F n ) ↓ µ(F ).
(⇒) Since µ is finite, by constant additivity it suffices to prove the statement for

f n , f ∈ B + ∞ (Ω). Let f n ↓ a.e.
f . There exists E with µ(E) = 0 such thatf n 1l E c ↓ e. f 1l E c . By the monotone convergence theorem for continuous from above set functions (see e.g. [START_REF] Rébillé | Sequentially continuous non-monotonic Choquet integrals[END_REF]) we have

f n 1l E c dµ ↓ f 1l E c dµ. Since µ(E) = 0 null-additivity entails gdµ = g1l E c dµ for g = f n , f . We get f n dµ ↓ f dµ. ⊓ ⊔
A way to construct strongly continuous from above set functions is to take: µ = f oP with P a (non-atomic) σ-additive probability and f :

[0, 1] -→ [0, 1], non-decreasing, f right-continuous with f (p) > 0 for p > 0.
Analogously we can provide some theorem with continuity from below. µ is said to be strongly continuous from below if for all E, F n such that µ(F ) = 0 and

F n ↓ F with F n ⊂ E then µ(E \ F n ) ↑ µ(E).
Proposition 3.2 Let µ be a set function. µ is strongly continuous from below ⇐⇒ µ is continuous from below and null-additive.

Proof: (⇒) To prove continuity from above, let A

n ↑ A. Set E = A, F n = A \ A n . We get A \ A n ↓ ∅ thus µ(A n ) = µ(E \ F n ) ↑ µ(E) = µ(A). For F n = F we get null-additivity. (⇐) Let E, F n be such that µ(F ) = 0 and F n ↓ F with F n ⊂ E. We have µ(E \ F n ) ↑ µ(E \ F ) by continuity and µ(E \ F ) = µ(F ) by null-additivity. ⊓ ⊔ Theorem 3.2 Let µ be a set function. Then, µ is strongly continuous from below ⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n ↑ a.e. f ⇒ f n dµ ↑ f dµ Proof: Monotonicity is immediate. (⇐) Let µ(F ) = 0 and F n ↓ F with F n ⊂ E. Then 1l E\Fn ↑ a.e. 1l E , thus µ(E\F n ) ↑ µ(E).
(⇒) Since µ is finite, by constant additivity it suffices to prove the statement for

f n , f ∈ B + ∞ (Ω). Let f n ↑ a.e. f . There exists E with µ(E) = 0 such thatf n 1l E c ↑ e. f 1l E c
. By the monotone convergence theorem for continuous from below set functions (see e.g. [START_REF] Rébillé | Sequentially continuous non-monotonic Choquet integrals[END_REF]) we have

f n 1l E c dµ ↑ f 1l E c dµ. Since µ(E) = 0 null-additivity entails gdµ = g1l E c dµ for g = f n , f . We get f n dµ ↑ f dµ. ⊓ ⊔
A combination of the previous theorems provides a Lebesgue's dominated convergence theorem.

g n = sup k:k≥n f k 1l N c , h n = inf k:k≥n f k 1l N c ,
we have g n ↓ a.e. f 1l N c , h n ↑ a.e. f 1l N c , applying Theorem 3.1, 3.2 gives us the conclusion. (⇐). We only check that µ is monotone. Take

f n = f = 1l A and g = 1l B with A ⊂ B. Then, µ(A) = f dµ ≤ g dµ = µ(B). ⊓ ⊔
Remark 1 Since we deal with bounded measurable functions the condition of domination [∀n, |f n | ≤ g] can be restated as

∃M > 0 / ∀n, |f n | ≤ M.1l Ω .

Almost uniform convergence

We will now extend Property 2.1 to monotone a.u. convergence. This type of convergence is related to the notion of autocontinuity which originates in Wang's 1984 founding article ( [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF]). As it will turn out, autocontinuity is related to a.u. convergence in the same manner than strong continuity is related to a.e. convergence.

Definition 3.1 µ is autocontinuous from above if for all E, F n , E ∩ F n = ∅, µ(F n ) -→ 0 ⇒ µ(E ∪ F n ) -→ µ(E). µ is autocontinuous from below if for all E, F n , F n ⊂ E, µ(F n ) -→ 0 ⇒ µ(E \ F n ) -→ µ(E).

Monotone autocontinuity from above

For our purpose we investigate a monotone version of autocontinuity one can guess already in Theorem 1 and 2 in [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF] (see also [START_REF] Takahashi | Conditions for convergence theorems in non-additive measure theory[END_REF]) and which is weaker than autocontinuity (see section 5),

Definition 3.2 µ is monotone autocontinuous from above if for all E, F n ↓, E ∩ F n = ∅, µ(F n ) ↓ 0 ⇒ µ(E ∪ F n ) ↓ µ(E).
By definition a monotone autocontinuous from above set function is monotone (take E = B, F 1 = A \ B, F n = ∅ for n ≥ 2) and null-additive (take F n = F and µ(F ) = 0).

We use the terminology of monotone autocontinuity from above, since we require that the sequence of F n 's must be non-increasing. This is a weaker version of strong continuity from above, for instance take µ additive but not σ-additive.

The relation between monotone autocontinuity from above and strong continuity from above can be made more precise (see also Proposition 5 in [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF], Theorem 5.8 in [START_REF] Wang | Fuzzy Measure Theory[END_REF], Theorem 3.2 in [START_REF] Pap | σ-null-additive set functions[END_REF]), Proposition 3.3 Let µ be a set function. Then, µ is monotone autocontinuous from above and order continuous ⇐⇒ µ is strongly continuous from above.

Proof:

(⇒). Let E, F n ↓ F with E ∩ F n = ∅ and µ(E) = 0. Since F n \ F ↓ ∅ we have µ(F n \ F ) ↓ 0 by order continuity. Monotone autoconti- nuity from above entails µ(F n ) = µ(F n \ F ∪ F ) ↓ µ(F ). Now by null-additivity, µ(E ∪ F n ) = µ(F n ) for all n, thus µ(E ∪ F n ) ↓ µ(F ). (⇐).
To obtain order continuity take E = F = ∅ and F n ↓ ∅.

To obtain monotone autocontinuity from above take E,

F n ↓, E ∩ F n = ∅, with µ(F n ) ↓ 0. Put F = ∩ n F n .
We have µ(F ) = 0 and so by continuity from above and null-additivity we get µ(E

∪ F n ) ↓ µ(E ∪ F ) = µ(E). ⊓ ⊔
We may state now Property 2.1 for monotone a.u. convergence.

Theorem 3.4 Let µ be a set function.

Then µ is monotone autocontinuous from above ⇐⇒

∀f n , f ∈ B ∞ (Ω), f n ↓ a.u. f ⇒ f n dµ ↓ f dµ Proof: Monotonicity is immediate. (⇐) Let E, F n ↓, µ(F n ) ↓ 0 and ǫ > 0. There exists N ǫ such that µ(F Nǫ ) < ǫ.
We have also for n ≥ N ǫ ,

1l E∪Fn\F Nǫ = 1l E\F Nǫ = 1l E∪F \F Nǫ so 1l E∪Fn ↓ a.u. 1l E and by hypothesis µ(E ∪ F n ) = 1l E∪Fn dµ ↓ 1l E dµ = µ(E).
(⇒) Let f n ↓ a.u. f . Without loss of generality we may assume that 0 ≤ f n ≤ 1 and that there exists a non-increasing sequence

E m ⊃ E m+1 with µ(E m ) < 1 m such that f n 1l E c m ↓ u f 1l E c m .
Let ǫ > 0 and m a positive integer. We have,

f n dµ ≤ f n 1l E c m + 1l Em dµ ≤ f 1l E c m + 1l Em + ǫ1l Ω dµ for n large enough since f n 1l E c m + 1l Em ↓ u f 1l E c m + 1l Em . So, lim n f n dµ ≤ f 1l E c m + 1l Em + ǫ1l Ω dµ
The right hand side gives,

f 1l E c m + 1l Em + ǫ1l Ω dµ = 1 0 µ({f 1l E c m + 1l Em > t}) dt + ǫµ(Ω) = 1 0 µ({f > t} ∩ E c m ∪ E m ) dt + ǫµ(Ω) = 1 0 µ({f > t} ∪ E m ) dt + ǫµ(Ω) and since µ({f > t} ∪ E m ) ↓ µ({f > t}) for all t ∈ (0, 1) and µ({f > t} ∪ E 1 ) ≤ µ(Ω) the monotone convergence theorem concludes that lim n f n dµ ≤ f dµ + ǫµ(Ω).

⊓ ⊔

Analogously we can provide a theorem with continuity from below.

Monotone autocontinuity from below

Definition 3.3 µ is monotone autocontinuous from below if for all E, F n ↓, F n ⊂ E, µ(F n ) ↓ 0 ⇒ µ(E \ F n ) ↑ µ(E).
The relation between monotone autocontinuity from below and strong continuity from below can be made more precise (see Proposition 5 in [START_REF] Wang | The autocontinuity of set function and the fuzzy integral[END_REF], Theorem 3.3 in [START_REF] Pap | σ-null-additive set functions[END_REF]), Proposition 3.4 Let µ be a set function. If µ is strongly continuous from below then µ is monotone autocontinuous from below. The converse holds if µ is order continuous.

Proof: Let F n ↓ F , F n ⊂ E with µ(F n ) ↓ 0. Hence µ(F ) = 0, thus µ(E \ F n ) ↑ µ(E).
For the converse. Let F n ↓ F , F n ⊂ E with µ(F ) = 0. By order continuity we have µ(F n \ F ) ↓ 0. And monotone autocontinuity from below entails µ((

E \ F ) \ (F n \ F )) ↑ µ(E \ F ) = µ(E), by null-additivity. ⊓ ⊔
We can note that the converse can hold if we do not assume order continuity.

Example 1 ([14]) Let P be a non-atomic σ-additive probability and consider the distortion function f defined by f (p) = 1 for p ∈ (0, 1], f (0) = 0. Let µ = f oP . We have

f dµ = ess P supf = inf{M : M ∈ IR such that P ({f > M }) = 0}.
This set function is strongly continuous from below, monotone autocontinuous from above, but is not order continuous.

Theorem 3.5 Let µ be a set function. Then, µ is monotone autocontinuous from below ⇐⇒

∀f n , f ∈ B ∞ (Ω), f n ↑ a.u. f ⇒ f n dµ ↑ f dµ Proof: Monotonicity is immediate. (⇐) Let E, F n ↓, µ(F n ) ↓ 0 and ǫ > 0.
There exists N ǫ such that µ(F Nǫ ) < ǫ.

We have also for n ≥ N ǫ , 1l (⇒) Let f n ↑ a.u. f . Without loss of generality we may assume that 0 ≤ f n ≤ 1 and that there exists a non-increasing sequence

E m ⊃ E m+1 with µ(E m ) < 1 m such that f n 1l E c m ↑ u f 1l E c m .
Assume there exists c > 0 such that c ≤ f 1 (ω) for all ω ∈ Ω. Let ǫ ∈ (0, c) and m a positive integer. We have,

f n dµ ≥ f n 1l E c m dµ ≥ f 1l E c m -ǫ1l E c m dµ for n large enough since f n 1l E c m ↑ u f 1l E c m . So, lim n f n dµ ≥ f 1l E c m -ǫ1l E c m dµ.
The right hand side gives,

f 1l E c m -ǫ1l E c m dµ = f 1l E c m dµ -ǫµ(E c m ), by comonotonicity ≥ f 1l E c m dµ -ǫµ(Ω) = 1 0 µ({f > t} \ E m ) dt -ǫµ(Ω)
and since µ({f > t} \ E m ) ↑ µ({f > t}) for all t ∈ (0, 1) the monotone convergence theorem concludes that lim n f n dµ ≥ f dµǫµ(Ω).

Now if inf f 1 = 0. Set ĝ = 1 2 (g + 1l Ω ) for g = f, f n .
We have 1 2 1l Ω ≤ fn ≤ 1l Ω and fn ↑ a.u. f . So lim n fn dµ ↑ f dµ and by comonotonic additivity we get lim n f n dµ ↑ f dµ. ⊓ ⊔

Monotone autocontinuity

A combination of the previous theorems gives us a Lebesgue's type dominated convergence theorem without sequential continuity.

Theorem 3.6 Let µ be a set function. Then, µ is monotone autocontinuous

⇐⇒ ∀f n , f, g ∈ B ∞ (Ω), |f n | ≤ g, f n a.u. -→ f ⇒ f n dµ → f dµ, f dµ ≤ gdµ Proof:
The same as in the classical case. Consider the bounded measurable functions

g n = sup k:k≥n f k , h n = inf k:k≥n f k .
We can check that g n , h n are ↓ a.u. , ↑ a.u. f and -g ≤ h n ≤ g n ≤ g. Since f n a.u.

-→ f there are

E m ⊃ E m+1 with µ(E m ) < 1 m such that f n 1l E c m u.
-→ f 1l E c m . Let p be a positive integer. There exists N p such that for all n ≥ N p and for all ω ∈ E c m , We can state a sufficient condition for an Egoroff's theorem ( [START_REF] Li | A further investigation for Egoroff's theorem with respect to monotone set functions[END_REF][START_REF] Murofushi | Conditions for Egoroff's theorem in non-additive measure theory[END_REF], see also [START_REF] Li | Order continuous of monotone set function and convergence of measurable functions sequence[END_REF] for a Lebesgue's Theorem).

f (ω) -ǫ ≤ f n (ω) ≤ f (ω) + ǫ thus f (ω) -ǫ ≤ h Np (ω) ≤ g Np (ω) ≤ f (ω) + ǫ. So h Np ↑ a.u. f , g Np ↓ a.u. f and h n ↑ a.u. f , g n ↓ a.u. f follow. ⊓ ⊔ Remark 2

Convergence in measure

A sequence {f n } n of bounded functions is said to converge strictly in measure to f denoted by

f n s-µ -→ f if lim n µ({|f n -f | > 0}) = 0.
The sµ convergence is stronger than convergence in measure which is defined as follows

f n µ -→ f if ∀ǫ > 0, lim n µ({|f n -f | ≥ ǫ}) = 0
Similarly convergence in measure is stronger than mean-convergence (see Theorem 2 in [START_REF] Wang | Convergence theorems for sequences of Choquet integrals[END_REF]), i.e.,

f n m -→ f if lim n |f n -f | µ = 0.
If we assume that the domination condition holds i.e.

|f n |, |f | ≤ g, ∈ B ∞ (Ω, A) then f n µ -→ f ⇒ f n m -→ f and a fortiori f n s-µ -→ f ⇒ f n m
-→ f . These sµ, m, µ convergences fully characterize autocontinuity. In particular, the equivalence (i) ⇐⇒ (ii) with sµ convergence can be an alternative to Theorem 3.6 with a.u. convergence. Theorem 3.7 Let µ be a monotone set function. The following assertions are equivalent, (i) µ is autocontinuous,

(ii) ∀f n , f, g ∈ B ∞ (Ω), |f n |, |f | ≤ g, f n s-µ -→ f ⇒ f n dµ -→ f dµ, (iii) ∀f n , f, g ∈ B ∞ (Ω), |f n |, |f | ≤ g, f n µ -→ f ⇒ f n dµ -→ f dµ, (iv) ∀f n , f, g ∈ B ∞ (Ω), |f n |, |f | ≤ g, f n m -→ f ⇒ f n dµ -→ f dµ.
The equivalence of (i), (iii) is a restatement of Theorem 6 and 7 in [START_REF] Wang | Convergence theorems for sequences of Choquet integrals[END_REF] where the condition of equiintegrability is dropped and the local uniform autocontinuity or the continuity conditions become too strong. Other formulations can be found as Theorem 3.5 in [START_REF] Dug | Convergence of Choquet Integral[END_REF] and as Theorem 3.3 in [START_REF] Murofushi | Conditions for Egoroff's theorem in non-additive measure theory[END_REF].

Proof: (ii) ⇒ (i). Let E, F n , E ∩ F n = ∅ and set f n = 1l E∪Fn , f = 1l E , g = 1l Ω . Assume µ(F n ) ↓ 0. That is, µ({|f n -f | > 0}) ↓ 0. We get µ(E∪F n ) = f n dµ -→ f dµ = µ(E). Similarly, let E, F n , F n ⊂ E with µ(F n ) ↓ 0 and set f n = 1l E\Fn , f = 1l E , g = 1l Ω . (iii) ⇒ (i), (iv) ⇒ (i). Same as (ii) ⇒ (i). (i) ⇒ (ii).
Since µ is assumed to be autocontinuous we can assume that |f n |, |f | ≤ g. By constant additivity we only consider the case where f n , f ≥ 0. Let t ∈ [0, M ] where M = sup g. We have,

{f n ≥ t} ⊂ {f ≥ t} ∪ {|f n -f | > 0} thus µ({f n ≥ t}) ≤ µ({f ≥ t} ∪ {|f n -f | > 0}) ≤ µ(Ω)
and by integration,

f n dµ ≤ [0,M ] µ({f ≥ t} ∪ {|f n -f | > 0}) dλ(t).
Now since µ is autocontinuous from above we also have

µ({f ≥ t} ∪ {|f n -f | > 0}) -→ µ({f ≥ t}),
thus by the dominated convergence theorem we get

lim f n dµ ≤ [0,M ] µ({f ≥ t}) dλ(t) = f dµ. Similarly, since {f ≥ t} \ {|f n -f | > 0} ⊂ {f n ≥ t} thus µ({f ≥ t} \ {|f n -f | > 0}) ≤ µ({f n ≥ t}) ≤ µ(Ω)
and by integration,

[0,M ] µ({f ≥ t} \ {|f n -f | > 0}) dλ(t) ≤ f n dµ.
Now since µ is autocontinuous from below we also have

µ({f ≥ t} \ {|f n -f | > 0}) -→ µ({f ≥ t}),
thus by the dominated convergence theorem we get

f dµ = [0,M ] µ({f ≥ t}) dλ(t) ≤ lim f n dµ.
Since ǫ is arbitrary, we obtain f dµ = lim n f n dµ.

(i) ⇒ (iii). The proof is adapted from [START_REF] Wang | Convergence theorems for sequences of Choquet integrals[END_REF]. Since µ is assumed to be autocontinuous we can assume that |f n |, |f | ≤ g. By constant additivity we only consider the case where f n , f ≥ 0. Let t ∈ [0, M ] where M = sup g and ǫ > 0. We have,

{f n ≥ t + ǫ} ⊂ {f ≥ t} ∪ {|f n -f | ≥ ǫ} thus µ({f n ≥ t + ǫ}) ≤ µ({f ≥ t} ∪ {|f n -f | ≥ ǫ}) ≤ µ(Ω)
and by integration,

f n dµ-ǫµ(Ω) ≤ [ǫ,M ] µ({f n ≥ t})dλ(t) ≤ [0,M ] µ({f ≥ t}∪{|f n -f | ≥ ǫ}) dλ(t).
Now since µ is autocontinuous from above we also have

µ({f ≥ t} ∪ {|f n -f | ≥ ǫ}) -→ µ({f ≥ t}),
thus by the dominated convergence theorem we get

lim f n dµ -ǫµ(Ω) ≤ [0,M ] µ({f ≥ t}) dλ(t) = f dµ. Similarly, since {f ≥ t} \ {|f n -f | ≥ ǫ} ⊂ {f n ≥ t -ǫ} thus µ({f ≥ t} \ {|f n -f | ≥ ǫ}) ≤ µ({f n ≥ t -ǫ}) ≤ µ(Ω)
and by integration,

[0,M ] µ({f ≥ t} \ {|f n -f | ≥ ǫ}) dλ(t) ≤ ǫµ(Ω) + [ǫ,M ] µ({f ≥ t} \ {|f n -f | ≥ ǫ}) dλ(t) ≤ ǫµ(Ω) + [ǫ,M ] µ({f n ≥ t -ǫ}) dλ(t) ≤ ǫµ(Ω) + [0,M ] µ({f n ≥ t}) dλ(t) = ǫµ(Ω) + f n dµ
Now since µ is autocontinuous from below we also have

µ({f ≥ t} \ {|f n -f | ≥ ǫ}) -→ µ({f ≥ t}),
thus by the dominated convergence theorem we get

f dµ = [0,M ] µ({f ≥ t}) dλ(t) ≤ lim f n dµ + ǫµ(Ω).
Since ǫ is arbitrary, we obtain f dµ = lim n f n dµ.

(iii) ⇒ (iv). Using Markov's inequality it holds, for all ǫ > 0,

ǫ.µ({|f n -f | ≥ ǫ}) ≤ |f n -f | dµ -→ 0.
We add a direct proof of (iv) ⇒ (iii). Let sup g = T < ∞. Then, for all t ∈ [0, T ],

µ({|f n -f | ≥ t}) ≤ µ({|f n | ≥ t/2} ∪ {|f | ≥ t/2}) ≤ µ({g ≥ t/2}) and T 0 µ({g ≥ t/2}) dt = 2 g dµ < ∞. Assume f n µ -→ f , then by the dominated convergence theorem applied to the decumulative function µ({|f n -f | ≥ (.)}) on [0, T ] it comes |f n -f | dµ = T 0 µ({|f n -f | ≥ t}) dt -→ 0 that is, f n m -→ f . ⊓ ⊔ Remark 3
The domination condition in Theorem 3.7 can be stated to hold almost everywhere (see Remark 2).

We may extract from the proof the following theorems, where sµ convergence can either be replaced by µ or m convergence, Theorem 3.9 Let µ be a monotone set function. Then, µ is autocontinuous from below ⇐⇒

∀f n , f, g ∈ B ∞ (Ω), [ |f n |, |f | ≤ g ] a.e., f n s-µ -→ f ⇒ f dµ ≤ lim f n dµ.
Strict convergence in measure is neither weaker nor stronger than a.e. convergence, a.u. convergence or m convergence. The domination condition is necessary to prove that µ convergence implies m convergence.

Example 2 Let µ be the Lebesgue measure on Ω = [0, 1) endowed with its Borel σ-algebra. Define the following sequences,

f n = 1 n 1l [0,1) , g n = 1l [ p 2 m , p+1 2
m ) for n = 1 + . . . + 2 m-1 + p, p = 0, . . . , 2 m -1,

h n = n1l (0, 1 n ) . Then, f n a.u. -→ 1l ∅ and f n s-µ → 1l ∅ g n s-µ -→ 1l ∅ and g n a.u. → 1l ∅ , g n a.e. → 1l ∅ h n s-µ -→ 1l ∅ and h n m → 1l ∅ .

Dual Results

Our previous results dealing with a.e. and a.u. convergence can be translated with pseudo convergence. A sequence {f n } n is said to pseudo everywhere converge to f if there is a set E with µ(E) = µ(Ω) such that {f n 1l E } n converges to f 1l E (see [START_REF] Wang | Asymptotic structural characteristics of fuzzy measure and their applications[END_REF]). Similarly, {f n } n is said to pseudo converge uniformly to f if for all ǫ > 0 there is a set E ǫ with µ(E ǫ ) > µ(Ω)ǫ such that {f n 1l Eǫ } n converges uniformly to f 1l Eǫ . Otherwise stated pseudo-convergence with respect to µ is convergence with respect to µ. For this reason we introduce the converse definition of the various continuity definition related to the conjugate set functions. A set function µ is said to be, pseudo strongly continuous from above if for all G,

H n ↑ H, H n ⊃ G, with µ(H) = µ(Ω) then µ(G ∪ H c n ) ↓ µ(G), pseudo strongly continuous from below if for all G, H n ↑ H, G ∪ H n = Ω, with µ(H) = µ(Ω) then µ(G ∩ H n ) ↑ µ(G), pseudo monotone autocontinuous from above if for all G, H n ↑, H n ⊃ G, µ(H n ) ↑ µ(Ω) then µ(G ∪ H c n ) ↓ µ(G), pseudo monotone autocontinuous from below if for all G, H n ↑, G ∪ H n = Ω, µ(H n ) ↑ µ(Ω) then µ(G ∩ H n ) ↑ µ(G).
These definitions are simply restating that µ is respectively strongly continuous from below, strongly continuous from above, monotone autocontinuous from below and monotone autocontinuous from above. We can now formulate the pseudo-convergence theorems.

ess P inff = sup{M : M ∈ IR such that P ({f > M }) = 1},
is continuous with respect to pseudo almost uniform convergence but not with respect to pseudo almost everywhere convergence.

Counterexamples

We shall present four examples of set functions which clarify the relationship between autocontinuity from above (below), monotone autocontinuity from above (below) and order continuity (see also [START_REF] Asahina | Relationship among continuity conditions and null-additivity conditions in non-additive measure theory[END_REF][START_REF] Takahashi | Conditions for convergence theorems in non-additive measure theory[END_REF][START_REF] Wang | On the null-additivity and the autocontinuity of a fuzzy measure[END_REF]). Examples 4 and 5 show that monotone autocontinuity and order continuity are independent properties. Example 5 show that monotone autocontinuity from above (below) is weaker than autocontinuity from above (below) even under order continuity. Examples 6 and 7 show that monotone autocontinuity from above and from below are independent even under order continuity. These examples are given on IN = {1, 2, 3, . . .} the set of positive integers. On IN, order continuity can be characterized2 in a simple way by the necessary condition lim 1/(4n) , if A = {n} for some n

1 2 min A , if |A| > 1 0 , if A = ∅.
Then, µ is neiher monotone autocontinuous from above nor from below but µ is order continuous. so µ is order continuous but µ is not monotone autocontinuous from above since

µ({1} ∪ [n, ∞)) = 1 -→ 1/2 = µ({1})
In order to prove monotone autocontinuity from below we shall prove the even stronger statement that µ is strongly continuous from below. By Proposition 3.2, it suffices to prove that µ is null-additive and continuous from below. Since µ(A) > 0, whenever A = ∅, µ is null-additive. Let us prove now that µ is continuous from below. Let {A n } n be an increasing sequence to A. We may assume that A is infinite. Put α = min A. Since {A n } n is increasing to A and α ∈ A thus there exists some n(A) such that α ∈ A n(A) , hence min A n(A) ≤ α.

But A n ⊂ A, so min A n ≥ α.

µ(A) =      1/ min A , if |A| = ∞ 1/(2 min A) , if |A| < ∞ 0 , if A = ∅.
Then, µ is monotone autocontinuous from above and order continuous but is not monotone autocontinuous from below and is not autocontinuous from above.

Proof: Let n > 1. We have, µ([n, ∞)) = 1/n -→ 0 so µ is order continuous. But µ is not monotone autocontinuous from below since

Theorem 3 . 3

 33 Let µ be a set function. Then µ is strongly continuous from below and above ⇐⇒ ∀f n , f, g ∈ B ∞ (Ω), |f n | ≤ g a.e., f n a.e. -→ f ⇒ f n dµ → f dµ, f dµ ≤ gdµ Proof: (⇒). We have µ({|f n | > g}) = 0 for all n. Thus by null-additivity µ(∪ n i=1 {|f i | > g}) = 0, and continuity from below entails µ(∪ n {|f n | > g}) = 0. Let N = ∪ n {|f n | > g}. Consider the bounded measurable functions

  E\Fn)\F Nǫ = 1l E\F Nǫ so 1l E\Fn ↑ a.u. 1l E and by hypothesis µ(E \ F n ) = 1l E\Fn dµ ↑ 1l E dµ = µ(E).

  The domination condition in Theorem 3.6 can be stated in a more general manner as, [ ∀n, |f n | ≤ g ] almost everywhere, i.e., µ(∪ n {|f n | > g}) = 0. If µ is σ-null-additive the domination condition reduces to for all n, [ |f n | ≤ g ] almost everywhere, i.e., ∀n, µ({|f n | > g}) = 0.

Corollary 3 . 1

 31 Let µ be a monotone autocontinuous set function. Then, µ is order continuous⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n a.e. -→ f ⇒ f n a.u.-→ f Proof: (⇒) From Propositions 3.3, 3.4 and Corollary 6.4 in[START_REF] Wang | Fuzzy Measure Theory[END_REF]. (⇐) Since µ is monotone autocontinuous, for any sequence {f n } n with f n a.e.-→ f and |f n |, |f | ≤ |g| we have f n dµ → f dµ, hence the sufficient part in Theorem 3.3 entails order continuity.⊓ ⊔

Theorem 3 . 8

 38 Let µ be a monotone set function. Then, µ is autocontinuous from above ⇐⇒∀f n , f, g ∈ B ∞ (Ω), [ |f n |, |f | ≤ g ] a.e., f n s-µ -→ f ⇒ lim f n dµ ≤ f dµ.

Corollary 4 . 1

 41 Let µ be a set function. Then, µ is pseudo strongly continuous from above ⇐⇒∀f n , f ∈ B ∞ (Ω), f n ↓ p.a.e. f ⇒ f n dµ ↓ f dµ Corollary 4.2 Let µ be a set function. Then, µ is pseudo strongly continuous from below ⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n ↑ p.a.e. f ⇒ f n dµ ↑ f dµCorollary 4.3 Let µ be a set function. Then, µ is pseudo monotone autocontinuous from above ⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n ↓ p.a.u. f ⇒ f n dµ ↓ f dµ Corollary 4.4 Let µ be a set function. Then, µ is pseudo monotone autocontinuous from below ⇐⇒ ∀f n , f ∈ B ∞ (Ω), f n ↑ p.a.u. f ⇒ f n dµ ↑ f dµ Proof: The proofs rely on the duality formula, i.e., f dµ = --f dµ, the fact that f n ↓ p.a. f ⇐⇒ -f n ↑ p.a. -f and that pseudo almost convergence with respect to µ is equivalent to almost convergence with respect to µ. ⊓ ⊔ Example 3 ([14]) The essential infimum functional, defined as

Example 4

 4 Let Ω = IN and A = 2 I N . Define,

ForExample 7

 7 n = α we obtain, min A n(A) = α. It follows that for n ≥ n(A), α = min A n(A) ≥ min A n ≥ α thus µ(A n ) = µ(A). ⊓ ⊔ Let Ω = IN and A = 2 I N . Define,

µ([ 1

 1 , n]) = 1/2 -→ 1 = µ(IN).And µ is not autocontinuous from above, sinceµ({1} ∪ [n, ∞)) = 1 -→ 1/2 = µ({1}). Let us prove that µ is monotone autocontinuous from above. Let E, F n ⊂ IN with F n ↓, µ(F n ) ↓ 0 and F n ⊂ E. Since F n is decreasing, each F n is infinite. As µ(F n ) ↓ 0, we have min F n ↑ ∞, thus F n ↓ ∅. Put e = min E. Since F n ↓ ∅ there exists n E such that for n ≥ n E , F n ∩ [1, e] = ∅,thus min F n > e. So we have for n ≥ n E , min E ∪ F n = min{min E, min F n } = e thus µ(E ∪ F n ) = µ(E). ⊓ ⊔

Indeed, let A n ↓ ∅. Then, min A n ↑ ∞. Thus, µ(A n ) ≤ µ([min A n , ∞)) ↓ 0.

Proof: Let n > 1. We have, µ([n, ∞)) = 1/(2n) -→ 0 so µ is order continuous. But,

so µ is not monotone autocontinuous from above and not monotone autocontinuous from below.

⊓ ⊔

Example 5 Let Ω = IN and A = 2 I N . Define,

Then, µ is monotone autocontinuous from above and below but µ is not order continuous and is neither autocontinuous from above nor from below.

Proof: µ is not order continuous since for all n, µ([n, ∞)) = 1/2.

We check now that µ is monotone autocontinuous from above and below. Let {F n } n be a decreasing sequence. Since each must F n is infinite, it holds for all n, µ(F n ) ≥ 1/2. Hence µ(F n ) -→ 0, thus µ is monotone autocontinuous. However, µ is not autocontinuous from above neither from below. Let n > 1.

We have,

Then, µ is not monotone autocontinuous from above but is monotone autocontinuous from below and order continuous.

Proof: Let n > 1. We have, µ([n, ∞)) = 1/n -→ 0