Is it possible to extend the set of measurable functions $F := \{ f: \mathbb{R} \to \mathbb{R} \mid f \text{ is Lebesgue measurable }\}$ to a Polish space?
I feel like the biggest obstacle is to find a metric on $F$ such that $F$ has a dense subset.
(Note that I do not want to restrict $F$ to continuous functions only.)
There are too many measurable functions for this to work: if we let $A$ be a measure-zero set, then the characteristic function of $A$ is measurable, but there are more than continuum many measure-zero sets - and every Polish space has size at most continuum.