ISSN 1064-5624, Doklady Mathematics, 2009, Vol. 80, No. 2, pp. 749–754. © Pleiades Publishing, Ltd., 2009. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 428, No. 6, pp. 727–732. 749 In works [1, 2], the problem has been addressed of approximating of the entropy functional on the space of measurable functions on a probability space (M, , µ). Below some assertions from these works are reinforced and precised. The entropy functional is defined by the formula The approximations are given by the formula S n ( f ) := S(A n ( f )), where {A n } is a sequence of operators with certain properties. In particular, one can consider the operators of conditional expectation with respect to some filtration. The question arises whether one can approximate in this way functionals of broader classes including the entropy. The entropy functional S has the form where V:  →  ∪ {∞} is a convex function (in the case of the entropy Below for convenience we consider such functionals with the opposite sign. We shall see that the result on approximation extends to the case of an arbitrary con- vex function V(·) and that one can consider an even broader class of functionals of the form Sf () f f d µ . ln M ∫ – = Sf () Vfx () ( )µ d x ( ) , M ∫ – = Vy () y y if y ln 0 ≥ +∞ if y 0 . < ⎩ ⎨ ⎧ = H V f () Vxfx () , ( )µ d x ( ) , f M ∫ L 1 µ ( ) , ∈ = where the function V is convex in the second variable and satisfies certain additional conditions with respect to the first variable. Moreover, we shall precise some assertions from [1]. Throughout the symbol ( f |) denotes the conditional expectation of f with respect to the σ-algebra . The facts from the theory of mea- sure and integral used below can be found in [3]. First we consider an example showing that in the presence of dependence of V on the first variable addi- tional assumptions are necessary indeed. Example 1. Let us take the interval [–1, 1] with the normalized Lebesgue measure. Let Then the function y  V(x, y) is convex for all x. Set f (x) = I [0, 1] (x). The sequence of decreasing partitions , k ≥ 1, of the interval [–1, 1] is defined in such a way that the partition with the index k con- tains the interval [–(k – 1)k –2 , k –2 ] and the remaining part is partitioned into k intervals in order to obtain a sequence of decreasing partitions of diameter δ k → 0. Denote by  k the σ-algebra generated by the sets , , …, . Then the sequence ( f | k ) converges to f almost everywhere, but Indeed, for any x ∈ I k := [–(k – 1)k –2 , k –2 ], we have . Vxy , ( ) 0 if x 1 – 0 , [ ] , ∈ y 3 – if x 01 , [ ] , y 01 ] , ( ∈ ∈ +∞ if x 01 , [ ] , y 1 – 0 , [ ] ∈ ∈ . ⎩ ⎪ ⎨ ⎪ ⎧ = Q i k { } i 12 … n k , , , = Q 1 k Q 2 k Q n k k Vxfx () , ( ) x d 1 – 1 , [ ] ∫ 1 2 - , = Vx  f  k ( ) x () , ( ) x d 1 – 1 , [ ] ∫ k ∞ → lim ∞. =  f  k ( ) x () k I 01 , [ ] x () x d I k ∫ 1 k - = = Approximation of Nonlinear Integral Functionals 1 V. I. Bogachev and A. A. Lipchyus Presented by Academician Yu.V. Prokhorov April 25, 2009 Received June 4, 2009 DOI: 10.1134/S1064562409050317 Faculty of Mechanics and Mathematics, Moscow State University, 119899 Russia e-mail: vibogach@mail.ru MATHEMATICS 1 The article was translated by the authors.