Σ-konečná míra

σ-konečná míra je v teorii míry označení takové míry, která je definována na σ-algebře $\Sigma$ tvořené podmnožinami množiny $X$ , přičemž platí, že $X$ lze vyjádřit jako spočetné sjednocení množin o konečné míře.

Formální definice

Nechť $(X,{\mathcal {A}})$ je měřitelný prostor s mírou $\mu$ . Pak se $\mu$ nazývá σ-konečná, pokud splňuje jednu z následujících čtyř ekvivalentních podmínek:

Množinu $X$ je možno pokrýt spočetnou množinou měřitelných množin o konečné míře. Tedy existují množiny $A_{1},A_{2},\ldots \in {\mathcal {A}}$ , kde $\mu \left(A_{n}\right)<\infty$ pro všechna $n\in \mathbb {N}$ a přitom $\bigcup _{n\in \mathbb {N} }A_{n}=X$
Množinu $X$ je možno pokrýt spočetnou množinou navzájem disjunktních množin o konečné míře. Tedy existují $B_{1},B_{2},\ldots \in {\mathcal {A}}$ , kde $\mu \left(B_{n}\right)<\infty$ a $n\in \mathbb {N}$ a $B_{i}\cap B_{j}=\varnothing$ pro $i\neq j$ , které splňují $\bigcup _{n\in \mathbb {N} }B_{n}=X$ .
Množinu $X$ je možno pokrýt monotónní posloupností měřitelných množin o konečné míře. Tedy existují množiny $C_{1},C_{2},\ldots \in {\mathcal {A}}$ s $C_{1}\subset C_{2}\subset \cdots$ splňující $\mu \left(C_{n}\right)<\infty$ pro všechna $n\in \mathbb {N}$ , přičemž platí $\bigcup _{n\in \mathbb {N} }C_{n}=X$ .
Existuje kladná měřitelná funkce $f$ , jejíž integrál je konečný, tedy: $f(x)>0$ pro všechna $x\in X$ a $\int f(x)\mu (\mathrm {d} x)<\infty$

Odkazy

Reference

V tomto článku byl použit překlad textu z článku σ-finite measure na anglické Wikipedii.

Literatura

LUKEŠ, Jaroslav; MALÝ, Jan. Míra a integrál. Praha: Karolinum, 2002. ISBN 80-246-0543-0. Kapitola A. Míra a měřitelné funkce.

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