Definition:Restricted Measure
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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
- Results about restricted measures can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $14$