3
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 $n >\frac {\int f dm} {m(E)}$.
1
They just mean a union of two disjoint (i.e. with empty intersection) sets. Ignore the definition in Wikipedia for this case.
1
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 ...
Only top voted, non community-wiki answers of a minimum length are eligible
Related Tags
measure-theory × 31332real-analysis × 10561
probability-theory × 5517
lebesgue-measure × 4430
lebesgue-integral × 3037
functional-analysis × 2423
probability × 2391
integration × 2182
analysis × 1970
lp-spaces × 1034
general-topology × 1006
solution-verification × 914
convergence-divergence × 882
stochastic-processes × 760
elementary-set-theory × 569
random-variables × 510
measurable-functions × 487
probability-distributions × 479
calculus × 430
reference-request × 415
geometric-measure-theory × 414
borel-sets × 388
ergodic-theory × 376
conditional-expectation × 320
outer-measure × 299