Definition:Restricted Measure

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$.

Then the restricted measure on $\Sigma'$ or the restriction of $\mu$ to $\Sigma'$ is the mapping $\nu: \Sigma \to \overline{\R}$ defined by:


 * $\forall E' \in \Sigma': \nu \left({E'}\right) = \mu \left({E'}\right)$

That is, $\nu$ is the restriction $\mu \restriction_{\Sigma'}$.

Also see

 * Restricted Measure is Measure
 * Restricting Measure Preserves Finiteness