Are wellbehaved functions $f: \mathbb R^\infty \to \mathbb {R^\infty}$ sometimes non-measurable?

Question

A function from $\mathbb R^n$ to $\mathbb R^n$ can only be non measurable if it is not well behaved (in the sense that no function that anyone typically uses in an applied setting will ever be non-measurable).

Is this also the case if $n=\infty$? Or do we have to worry about possible non-measurability there?