Strong supmeasurability of functions of two variables whose vertical sections are preponderantly continuous (Q2464602)

Let \((X,\mathcal M)\) be a measurable space, \(\mathcal T\) a \(\sigma-\)ideal in \(\mathcal M\), \((Y,\mathcal N,\nu)\) a measure space and \(Z\) be a Banach space. A function of the form \(\sum_{n=1}^\infty \chi_{A_n}z_n\) with \(A_n\in\mathcal M\otimes\mathcal N\) and \(z_n\in Z\) is called nearly simple. A function \(f:X\times Y\rightarrow Z\) is called strongly measurable if there is a sequence of nearly simple functions \(f_n:X\times Y\rightarrow Z\) and a set \(A\in \mathcal T\) such that \((f_n)\) converges to \(f\) pointwise on \((X\setminus A)\times Y\). For any \(n\in\mathbb{N}\), let \({\mathcal J}_n\) be a countable cover of \(Y\) consisting of pairwise disjoint measurable sets of positive finite measure and \({\mathcal J}:=\bigcup_{n=1}^\infty{\mathcal J}_n\). \(\varphi:X\rightarrow Y\) is \((\mathcal M,\mathcal J)-\)measurable if \(\varphi^{-1}(J)\in\mathcal M\) for any \(J\in\mathcal J\). A function \(f:X\times Y\rightarrow Z\) is strongly supmeasurable if for each \((\mathcal M,\mathcal J)-\)measurable function \(\varphi:X\rightarrow Y\) the superposition \(x\mapsto f(x,\varphi(x))\) is strongly measurable. A function \(g:Y\rightarrow Z\) is called preponderantly continuous if \(g\) is measurable and for each \(y\in Y\) and each open neighborhood \(U\) of \(g(y)\) one has \(\nu(g^{-1}(U)\cap J_n(y))/\nu (J_n(y)) >1/2\) eventually where \(J_n(y)\) denotes the unique set in \({\mathcal J}_n\) containing \(y\). The main result says: Assume that \((\mathcal M,\mathcal T)\) satisfies the countable chain condition. Let \(f:X\times Y\rightarrow Z\) be a strongly measurable function with the property that \(\mathcal T-\)almost all vertical sections \(y\mapsto f(x,y)\) of \(f\) are preponderantly continuous. Then \(f\) is strongly supmeasurable.