Definition:Restriction of Measure to Trace Sigma-Algebra of Measurable Set

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure on $\struct {X, \Sigma}$.

Let $A \in \Sigma$.

Let $\Sigma_A$ be the trace $\sigma$-algebra of $A$ in $\Sigma$.

We define the restriction of $\mu$ to $\Sigma_A$, written $\mu \restriction_{\Sigma_A}$, as the restriction of $\mu$ to $\Sigma_A$ as a mapping.

Also see

• Trace Sigma-Algebra of Measurable Set shows that $\Sigma_A \subseteq \Sigma$ so the restriction is properly understood
• Restriction of Measure to Trace Sigma-Algebra of Measurable Set is Measure shows that $\mu \restriction_A$ is a measure on $\struct {A, \Sigma_A}$
• Definition:Restricted Measure - a more general concept that only allows restrictions to sub-$\sigma$-algebras of $\Sigma$, while $\Sigma_A$ will omit $X$ for $A \ne X$.