Definition:Intersection Measure

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $F \in \Sigma$.

Then the intersection measure (of $\mu$ by $F$) is the mapping $\mu_F: \Sigma \to \overline \R$, defined by:

$\map {\mu_F} E = \map \mu {E \cap F}$

for each $E \in \Sigma$.

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Signed Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

Let $F \in \Sigma$.

Then the intersection (signed) measure (of $\mu$ by $F$) is the mapping $\mu_F: \Sigma \to \overline \R$, defined by:

$\map {\mu_F} E = \map \mu {E \cap F}$

for each $E \in \Sigma$.

Complex Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.

Let $F \in \Sigma$.

Then the intersection (complex) measure (of $\mu$ by $F$) is the mapping $\mu_F: \Sigma \to \C$, defined by:

$\map {\mu_F} E = \map \mu {E \cap F}$

for each $E \in \Sigma$.

Also see

• Intersection Measure is Measure shows that $\mu_F$ is indeed a measure on $\struct {X, \Sigma}$.

• Results about intersection measures can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $7$