# Equivalence of definitions of measurable function: preimage of open set vs preimage of Borel set?

Rudin defines a function from a measure space $X$ to a topological space $Y$ as measurable if the preimage of every open set in $Y$ is measurable in the measure space $X$.

Now Wikipedia, and my books on probability, defines a function between two measure spaces $X$ and $Y$ as measurable if the preimage of every Borel set in $Y$ is measurable in $X$.

Since the inverse image preserves set operations, one can easily see that the set of subsets of $Y$ such that $f^{-1 }$ is measurable in $X$ forms a $\sigma$ algebra $\Sigma$ in $Y$. This implies $\Sigma \supset \mathcal B$, since $\Sigma$ contains all the opens sets.

Can one show the opposite inclusion, so that the defintions are in fact equal?