Strictly localizable measures
Strictly localizable measures [ edit ] Main article: Decomposable measure Semifinite measures [ edit ] Let � be a set, let � be a sigma-algebra on � , and let � be a measure on � . We say � is semifinite to mean that for all � ∈ � pre { + ∞ } , � ( � ) ∩ � pre ( � > 0 ) ≠ ∅ . [2] Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.) Basic examples [ edit ] Every sigma-finite measure is semifinite. Assume � = � ( � ) , let � : � → [ 0 , + ∞ ] , and assume � ( � ) = ∑ � ∈ � � ( � ) for all � ⊆ � . We have that � is sigma-finite if and only if � ( � ) < + ∞ for all � ...