Consider $(\Omega, \mathcal{F})$ and $(E,\mathcal{B}(E))$ two measurable spaces where $\mathcal{B}(E)$ is a Borel sigma algebra and $E$ is a metric space. We have a function $Y$ from $\Omega$ to $E$. Is it possible to define $Y$ as a continuous function without more information on the structure of $\Omega$ ?
My point here is that I have seen that there is a kind of "analogy" between measurable functions and continous functions, I see how it can be true when considering a function from $\mathbb{R}$ to $\mathbb{R}$ however when I try to take more "abstract" space I don't see how we can use this analogy without assuming a metric structure on $\Omega$.
Thank you
In order to define a continuous function you must have at least a topology on $\Omega$. Sometimes a measure is defined on the Borel field, which implicitly assumes a topological structure, but in the general abstract case, a $\sigma$-algebra on which a measure is defined need not have a topological structure.