Definition:Restricted Measure

Definition
Let $\struct {X, \Sigma, \mu}$ 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:


 * $\forall E' \in \Sigma': \map \nu {E'} = \map \mu {E'}$

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

Also see

 * Restricted Measure is Measure
 * Restricting Measure Preserves Finiteness