Let (u in mathcal{L}^{2}(mu)) be a positive, integrable function. Show that (int_{{u>1}} u d mu leqslant mu{u>1})

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Let \(u \in \mathcal{L}^{2}(\mu)\) be a positive, integrable function. Show that \(\int_{\{u>1\}} u d \mu \leqslant \mu\{u>1\}\) entails that \(u(x) \leqslant 1\) for \(\mu\)-a.e. \(x\).

Remark. This argument is needed for the second part of the proof of Lemma 27.4 (x).

Data from lemma 27.4 (x)

(x) 0

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