In The strength of measurability hypotheses, Raisonnier and Stern formulated the following principle:
(MUP). For any family $(A_x)_{x\in B}$ of non-empty subsets of$2^\omega$, indexed by the elements of a set $B$ of positive measure,there is a Borel function $f$, such that $\{x\in B\mid f(x)\notin A_x\}$ has Lebesgue measure zero.
They proceed to note that MUP implies the measurability of all sets. The argument: "in order to prove the measurability of $A$, one just uniformizes its characteristic function."
My question is: what does this mean? The characteristic function $\chi_A$ of $A$ is a subset of $(2^\omega)^2$, but I don't see how uniformizing it can get us anywhere (or needs the MUP to begin with). So it would seem I am missing something obvious.