Let $(X,\mu)$ be a standard probability space, and $(Y,\tau)$ an uncountable Polish space. Then the set $L^0(X,\mu,Y)$ of measurable maps from $X$ to $Y$ identified up to measure 0 is Polish w.r.t. the topology of convergence in measure.

It is then easy to see that the subset of maps $X\to Y$ with countable range is analytic, but is it Borel?