Definition:Restricted Measure

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

Let $\mathcal B$ be a sub-$\sigma$-algebra of $\mathcal A$.

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


 * $\forall B \in \mathcal B: \nu \left({B}\right) = \mu \left({B}\right)$

That is, $\nu$ is the restriction $\mu \restriction_{\mathcal B}$.

Also see

 * Restricted Measure is Measure
 * Restricting Measure Preserves Finiteness