How is defined $\mu|_A$?

Question

Let $(S,\Sigma,\mu)$ be a measurable space and $A\in\Sigma$. I can't find the definition for functions $\mu|_A$ and for $f|_A$ if $f \in m\Sigma$ (is measurable).

Answer 1

If we have $f:S\to\mathbb R$ then the restriction to $A\subseteq S$ is defined as $$f\vert_A : A\to \mathbb R,\quad x\mapsto f(x),\quad x\in A$$

It's a bit strange to write $\mu \vert _A$ since $\mu :\Sigma \to [0,\infty ]$. Perhaps they mean $\mu \vert _{\Sigma \cap A}$. Perhaps, it's just notation to denote something wholly different, entirely. Good lecture notes always specify these notations somewhere.