Restriction of a measure

Remarks

The map $\mu|_A$ is a measure on $(X, \Sigma)$, because intersecting a disjoint family with $A$ gives a disjoint family with the same union intersected with $A$. Example: if $\lambda$ is Lebesgue measure on $\R$ and $A = [0, 1]$, then $\lambda|_A([0, 5]) = \lambda([0, 1]) = 1$. For a measurable $f \colon X \to [0, \infty]$ one has $\int_X f \, d(\mu|_A) = \int_A f \, d\mu$. Non-example: for a set $A \notin \Sigma$, the sets $A \cap B$ need not lie in $\Sigma$, so the formula need not define a measure on $\Sigma$. A related view keeps only the points of $A$: the trace $\sigma$-algebra $\Sigma_A = \set{A \cap B : B \in \Sigma}$ on $A$, with the measure $\mu$ restricted to $\Sigma_A$. Other notations are $\mu \llcorner A$ and $\mu(\,\cdot \cap A)$.