The comments in the post

has led me to the following question:

Suppose that $(\Omega,\mathcal{F},P)$ is a probability space. Denote by $\mathcal{B}(\mathbb R)$ the Borel sigma-algebra on $\mathbb R$.

Let $A\in\mathcal{F}\otimes\mathcal{B}(\mathbb{R})$ be such that $P\otimes\lambda(A)>0$.

Is it true that there exists a $\mathcal{F}$-measurable random variable $X\colon \Omega\to\mathbb R$ such that $$ P(\{\omega\colon(\omega,X(\omega))\in A\})>0? $$ If it does not true, under which conditions on $(\Omega,\mathcal F,P)$ it holds?