Proposition 15.5.2. Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, $f, g: X \to Y$ be $(\cm, \cb_{Y})$-measurable functions, then the following functions are measurable:

1. For any metric $d$ on $Y$, $x \mapsto d(f(x), f(y))$.

2. If $Y$ is a TVS over $K \in \RC$ and $\lambda \in K$, $\lambda f + g$. In particular, $\bracs{f = g}\in \cm$.

3. If $Y \in \RC$, $fg$.