σ-finite measure - Wikipedia

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In mathematics, given a positive or a signed measure 𝜇 on a measurable space ( 𝑋 , 𝐹 ) , a 𝜎 -finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure 𝜇 is called a 𝜎 -finite measure if the set 𝑋 is 𝜎 -finite. A finite measure, for instance a probability measure, is always 𝜎 -finite. A different but related notion that should not be confused with 𝜎 -finiteness is s-finiteness. Let ( 𝑋 , 𝐴 ) be a measurable space and 𝜇 a measure on it. The measure 𝜇 is called a σ-finite measure, if it satisfies one of the four following equivalent criteria: If 𝜇 is a 𝜎 -finite measure, the measure space ( 𝑋 , 𝐴 , 𝜇 ) is called a 𝜎 -finite measure space.[3] If ( 𝑋 , 𝐴 , 𝜇 ) is a probability space, then the probability measure, 𝜇 is σ-finite, because 𝑋 is trivially covered by itself: 𝜇 ( 𝑋 ) = 1. For example, Lebesgue measure on the real numbers is not finite, but it is σ-fin

σ-finite measure - Wikipedia Jump to content From Wikipedia, the free encyclopedia Concept in measure theory 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 proba

σ-finite measure - Wikipedia

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