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 $n >\frac {\int f dm} {m(E)}$.


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


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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