Definition

Let $(X,\mathscr{M},\mu)$ be a measure space, and let $f:X\to \mathbb{R}$ be a $\mathscr{M}$-measurable function. Consider $\mathscr{N}=\{ E\subseteq \mathbb{R}:f^{-1}(E)\in\mathscr{M} \}$. We say that $\mathscr{N}$ is a σ-algebra containing all Borel subsets of $\mathbb{R}$.