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单词 MeasurabilityOfAnalyticSets
释义

measurability of analytic sets


AnalyticPlanetmathPlanetmath subsets (http://planetmath.org/AnalyticSet2) of a measurable spaceMathworldPlanetmathPlanetmath (X,) do not, in general, have to be measurable. See, for example, a Lebesgue measurable but non-Borel set (http://planetmath.org/ALebesgueMeasurableButNonBorelSet). However, the following result is true.

Theorem.

All analytic subsets of a measurable space are universally measurable.

Therefore for a universally complete measurable space (X,) all -analytic setsMathworldPlanetmath are themselves in and, in particular, this applies to any completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath σ-finite measure space (http://planetmath.org/SigmaFinite) (X,,μ). For example, analytic subsets of the real numbers are Lebesgue measurable.

The proof of the theorem follows as a consequence of the capacitability theorem.Suppose that A is an -analytic set. Then, for any finite measureMathworldPlanetmath μ on (X,), let μ* be the outer measure generated by μ. This is an -capacity and, by the capacitability theorem, A is (,μ*)-capacitable, hence is in the completion of with respect to μ (see capacity generated by a measure (http://planetmath.org/CapacityGeneratedByAMeasure)). As this is true for all such finite measures, A is universally measurable.

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更新时间:2025/5/4 6:54:19