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In line 3 you should say 'Let $f$ be a Lebegue integrable function on $\mathbb R$'. The inequality holds for all $n$. If $m(E)>0$ you get a contradiction by chosing an integer $n >\frac {\int f dm} {m(E)}$.


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They just mean a union of two disjoint (i.e. with empty intersection) sets. Ignore the definition in Wikipedia for this case.


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To prove the monotonicity property, the author has made use of the finite additivity property and the fact that measures are non-negative. More precisely, given a collection of pairwise disjoint sets $A_{k}\in\mathcal{F}$, where $1\leq k\leq n$, one has that \begin{align*} \mu\left(\bigcup_{k=1}^{n}A_{k}\right) = \sum_{k=1}^{n}\mu(A_{k}) \end{align*} At your ...


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