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单词 ExampleOfFunctionNotLebesgueMeasurableWithMeasurableLevelSets
释义

example of function not Lebesgue Measurable with measurable level sets


Consider V as in Vitali’s theorem. Define the function f:[0,1][0,+[ by:

f(x)={xifxV2+xifxV

The level sets of f will either be the empty setMathworldPlanetmath, or a singleton and thus measurable.

f-1({x})={{x}if 0x1xV{2-x}if 2x3x-2V{}otherwise

f is not a measurable functionMathworldPlanetmath since f-1([2,+[)=V and V is not a measurable setMathworldPlanetmath.

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更新时间:2025/5/5 3:43:14