This question assumes the existence of a large real-valued measurable cardinal. Let $X$ be an uncountable set and $(X,2^X)$ equipped with a non-atomic probability distribution $P$. Additionally, let $Q$ be the "uniform" distribution on the set $\{0,1\}^X$ --- i.e., the product of independent Bernoulli(1/2) random variables indexed by $X$ --- whose existence follows from the Łomnicki-Ulam theorem. The $\sigma$-algebra for the latter probability space is the usual cylindrical one used for random processes.

Let us now draw a "random function" $h\sim Q$ and a random point $x\sim P$.

Question: Is the object $h(x)$ a measurable random variable on the corresponding product space?