- Locally finite measure
In
mathematics , a locally finite measure is a measure for which every point of themeasure space has a neighbourhood offinite measure.Definition
Let ("X", "T") be a Hausdorff
topological space and let Σ be a σ-algebra on "X" that contains the topology "T" (so that everyopen set is ameasurable set , and Σ is at least as fine as the Borel σ-algebra on "X"). A measure/signed measure /complex measure "μ" defined on Σ is called locally finite if, for every point "p" of the space "X", there is an open neighbourhood "N""p" of "p" such that the "μ"-measure of "N""p" is finite.In more condensed notation, "μ" is locally finite
if and only if :forall p in X, exists N_{p} in T mbox{ s.t. } p in N_{p} mbox{ and } left| mu (N_{p}) ight| < + infty.
Examples
# Any
probability measure on "X" is locally finite, since it assigns unit measure the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite.
#Lebesgue measure onEuclidean space is locally finite.
# By definition, anyRadon measure is locally finite.
#Counting measure is sometimes locally finite and sometimes not: counting measure on theinteger s with their usualdiscrete topology is locally finite, but counting measure on thereal line with its usual Borel topology is not.ee also
*
Inner regular measure
*Strictly positive measure References
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