Theorem (1.12)

Let $(X,\mathscr{M})$ be a Measurable Space, let $(Y,\mathscr{T}_{Y})$ be a Topological Space, and $f:X\to Y$ a function.

1. Let $$\mathscr{N}=\{ E\subseteq Y:f^{-1}(E)\in \mathscr{M} \}.$$Then $\mathscr{N}$ is a σ-algebra of subsets of $Y$.
2. If $E\subseteq Y$ is a Borel set and $f$ is $\mathscr{M}$-measurable, then $$f^{-1}(E)\in\mathscr{M}$$
3. If $(Z,\mathscr{T}_{Z})$ is a Topological Space and $g:Y\to Z$ is a Borel function, then $$g\circ f:X\to Z$$is $\mathscr{M}$-measurable.