Functions of Two Variables Whose Vertical Sections Are Equiderivatives

We examine functions of two variables whose all vertical sections are equiderivatives. In particular we show that a bounded function whose horizontal sections are strongly measurable and vertical sections are equiderivatives, is strongly measurable. The theorems we prove are generalizations of the results of Z. Grande [3]. Let R and N denote the real line and the set of positive integers, respectively. Let (X,M) be a measurable space and let I ⊂M be a proper σ-ideal of subsets of X. Assume that Z is a Banach space. Let h : X → Z. Recall that h is measurable, if h−1(U) ∈ M for every open set U ⊂ Z. In [2], I introduced the following two kinds of measurability of a function. We say that h is nearly simple, if there exist elements α1, α2, . . . ∈ Z and a sequence of pairwise disjoint sets A1, A2, . . . ∈M such that h = αn on An for each n, and X = ⋃∞ n=1An. We say that h is strongly measurable with respect to (M, I), if there exists a sequence of nearly simple functions (hn) and a set A ∈ I such that hn → h on X \A. 2000 Mathematics Subject Classification. 26B05, 26B99, 26A15.