Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f : X \to Y$. Is $M(X,Y)$ also standard Borel?
First of all, the cardinality of $M(X,Y)$ is $\mathfrak{c} = 2^{\aleph_0}$ for uncountable $X$ and $Y$ (see Cardinality of the borel measurable functions?) - so this doesn't contradict the Borel ismorphism theorem.
In Srivastava, "A course on Borel sets", he considers the space of $B(X,Y) \subseteq M(X,Y)$ of Baire functions, i.e. continuous functions and closed under pointwise limit. Then he states the Lebesgue – Hausdorff theorem that $B(X,Y) = M(X,Y)$ for metrizable $X$. But I haven't found a theorem or note in the book that says that $B(X,Y)$ is standard Borel.
Moreover, he also states that any Borel measurable function can be made continuous by taking a finer topology on $X$ that doesn't change the Borel $\sigma$-algebra of $X$, i.e. $X$ is still standard Borel. But I don't see, how to apply this theorem.
