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history-page39 - σ is negligible ◦ Lebesgue ⋆ A set...

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Characterizations of Riemann-integrability Riemann The norm of a partition is the width of its largest subinterval. A function f is integrable if and only if, for any σ > 0 no matter how small, we can find a norm so that, for all partitions of [a, b] having a norm that small or smaller, the total length of the subintervals where the function oscillates more than
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Unformatted text preview: σ is negligible. ◦ Lebesgue ⋆ A set has measure zero if it “can be enclosed in a Fnite or a denumerable inFnitude of intervals whose total length is as small as we wish”. Examples of sets of measure zero are N and Q . ⋆ A bounded function is integrable if and only if the set of its points of discontinuity has measure zero....
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