Calculus Small Changes, Big Ideas

Section 24.5 Lebesgue measure

• The length of an interval \([a,b]\) is \(b-a\) and is denoted \(\cl L[a,b]\text{.}\) This is also the length of \((a,b)\text{,}\) \((a,b]\text{,}\) and \([a,b)\text{.}\) Intervals that involve \(\pm\infty\) are said to have length \(\infty\text{.}\)

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• A set \(A\) has measure zero if for all \(\eps>0\) there exists a countable collection \(I_{1},I_{2},I_{3},\cdots\) of intervals such that

  \begin{equation*} A\subseteq\bigcup_{k=1}^{\infty} I_{k}\quad\text{and}\quad\sum_{k=1}^{\infty}\cl L(I_{k})<\eps\text{.} \end{equation*}

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• Lebesgue’s theorem: Assume \(f:[a,b]\to\bb R\) is a bounded function and let \(\cl D\) be the set of points at which \(f\) is discontinuous. Then \(f\) is integrable if and only if \(\cl D\) has measure zero.

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• Go into the Lebesgue integral

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