Is it true in general that if [itex]f[/itex] is Lebesgue integrable in a measure space [itex](X,\mathcal M,\mu)[/itex] with [itex]\mu[/itex] a positive measure, [itex]\mu(X) = 1[/itex], and [itex]E \in \mathcal M[/itex] satisfies [itex]\mu(E) = 0[/itex], then(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int_E f d\mu = 0

[/tex]

This is one of those things I "knew" to be true yesterday, and the day before, and the day before...but now I can't show it! I need to be able to bound that integral, somehow, by [itex]\mu(E)[/itex], but how? Using Holder's inequality? But don't I need to know that [itex]f\in L^2[/itex] or [itex]f\in L^\infty[/itex] to do that? Do I know either of those? I don't think so...

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# Lebesgue integration over sets of measure zero

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