# Continuous Mapping is Measurable

## Theorem

Let $\left({X, \tau}\right)$ and $\left({X', \tau'}\right)$ be topological spaces.

Denote with $\mathcal B \left({X, \tau}\right)$ and $\mathcal B \left({X', \tau'}\right)$ their respective Borel $\sigma$-algebras.

Let $f: X \to X'$ be a continuous mapping.

## Proof

As $f$ is a continuous mapping, by definition, it holds that:

$\forall S' \in \tau': f^{-1} \left({S'}\right) \in \tau$

Now, by definition, $\mathcal B \left({X, \tau}\right) = \sigma \left({\tau}\right)$, and so $\tau \subseteq \mathcal B \left({X, \tau}\right)$.

Also, $\mathcal B \left({X', \tau'}\right) = \sigma \left({\tau'}\right)$.

Hence Mapping Measurable iff Measurable on Generator applies.

$\blacksquare$