Show that $E =\{(x,\alpha )\mid|f(x)|>\alpha\geq 0 \}$ is measurable.
113778
Show that $E=\{(x,\alpha)\mid 0\leq \alpha<|f(x)|\}$ is measurable if $f$ is measurable.
122585
Show that a function almost everywhere continuous is measurable
38324
Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$
98200
$E$ Lebesgue measurable implies $E^2$ Lebesgue measurable?
158122
If $f$ is measurable, then support of $f$ is measurable?
83825
$f^{-1}([-\infty,r))$ measurable for all rational numbers $r$
61133
Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
143699
Bounded measurable function and integral with charcteristic function
140790
Prove that $E(X) = \int_{0}^{\infty} P(X>x)\,dx = \int_{0}^{\infty} (1-F_X(x))\,dx$.
6450
Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
115219
If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<\epsilon$
22597
Composition of measurable function and continuous function
64486
Show that for non-negative measurable functions $f,g$ with $fg \geq 1$ the inequality $(\int f^p)(\int g^p) \geq 1$ holds.
146707
Formally show that the set of continuous functions is not measurable
121504
Supremum of a product measurable function...
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$