In mathematics, given a positive or a signed measure μ {\displaystyle \mu } on a measurable space ( X , F ) {\displaystyle (X,{\mathcal {F}})} , a σ {\displaystyle \sigma } -finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure μ {\displaystyle \mu } is called a σ {\displaystyle \sigma } -finite measure if the set X {\displaystyle X} is σ {\displaystyle \sigma } -finite. A finite measure, for instance a probability measure, is always σ {\displaystyle \sigma } -finite. A different but related notion that should not be confused with σ {\displaystyle \sigma } -finiteness is s-finiteness.
Definition Let ( X , A ) {\displaystyle (X,{\mathcal {A}})} be a measurable space and μ {\displaystyle \mu } a measure on it. The measure μ {\displaystyle \mu } is called a σ-finite measure, if it satisfies one of the four following equivalent criteria:
the set X {\displaystyle X} can be covered with at most countably many measurable sets with finite measure. This means that there are sets A 1 , A 2 , … ∈ A {\displaystyle A_{1},A_{2},\ldots \in {\mathcal {A}}} with μ ( A n ) < ∞ {\displaystyle \mu \left(A_{n}\right)<\infty } for all n ∈ N {\displaystyle n\in \mathbb {N} } that satisfy ⋃ n ∈ N A n = X {\displaystyle \bigcup _{n\in \mathbb {N} }A_{n}=X} . the set X {\displaystyle X} can be covered with at most countably many measurable disjoint sets with finite measure. This means that there are sets B 1 , B 2 , … ∈ A {\displaystyle B_{1},B_{2},\ldots \in {\mathcal {A}}} with μ ( B n ) < ∞ {\displaystyle \mu \left(B_{n}\right)<\infty } for all n ∈ N {\displaystyle n\in \mathbb {N} } and B i ∩ B j = ∅ {\displaystyle B_{i}\cap B_{j}=\varnothing } for i ≠ j {\displaystyle i\neq j} that satisfy ⋃ n ∈ N B n = X . {\displaystyle \bigcup _{n\in \mathbb {N} }B_{n}=X.}
the set X {\displaystyle X} can be covered with a monotone sequence of measurable sets with finite measure. This means that there are sets C 1 , C 2 , … ∈ A {\displaystyle C_{1},C_{2},\ldots \in {\mathcal {A}}} with C 1 ⊂ C 2 ⊂ ⋯ {\displaystyle C_{1}\subset C_{2}\subset \cdots } and μ ( C n ) < ∞ {\displaystyle \mu \left(C_{n}\right)<\infty } for all n ∈ N {\displaystyle n\in \mathbb {N} } that satisfy ⋃ n ∈ N C n = X {\displaystyle \bigcup _{n\in \mathbb {N} }C_{n}=X} . there exists a strictly positive measurable function f {\displaystyle f} whose integral is finite. This means that f ( x ) > 0 {\displaystyle f(x)>0} for all x ∈ X {\displaystyle x\in X} and ∫ f ( x ) μ ( d x ) < ∞ . {\displaystyle \int f(x)\mu (\mathrm {d} x)<\infty .}
… excerpt ends here. Continue reading the full article.
