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    Absolute continuity of the integral of an $L^1$ function

    PR-O4AY4

    $L^1$ functions have arbitrarily small integral on sets of sufficiently small measure.Suggest a slogan

    Let \(m\) be Lebesgue measure on \(\RR^n\) and let \(f\in L^1(\RR^n)\). For every \(\varepsilon>0\) there exists \(\delta>0\) such that every Lebesgue measurable set \(E\subseteq\RR^n\) with \(m(E)<\delta\) satisfies \[ \abs{\int_E f}\leq\int_E \abs{f} < \varepsilon . \]