Definition (σ-algebra containing all Borel subsets)

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 $\mathbb{R}$.

Definition (Law)

Let $(X,\mathscr{M},\mu)$ be a Measure Space, and let $f:X\to \mathbb{R}$ be a $\mathscr{M}$-Measurable Function. Let $\mathscr{N}$ be the of $\mathbb{R}$. Now define $\mu_{f}:\mathscr{N}\to[0,+\infty]$ as $$\mu_{f}(E):=\mu(f^{-1}(E))\quad E\in\mathscr{N}$$Then $\mu_{f}$ is a Measure and it is called the law of $f$.