The bookUnit 1 · Lebesgue Measure and Integration18 / 83

Measurable sets and functions

Why this is asked: Measure zero, countable additivity and 'almost everywhere' are the workhorses. Know that measurable ⊋ Borel and that a non-measurable set requires the axiom of choice.

Lebesgue measure and measurable functions

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A set with no measure: the Vitali constructioninteractive

Why Lebesgue measure cannot be defined on every subset of ℝ — built in four steps.

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Let X be a non-empty finite set and Y = { $f^{-1}(0)$ : f is a real-valued function on X}. Which one of the following statements is true?

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