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

• Results about restricted measures can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $14$