$\int_X f^p d\mu = p\int_{[0,+\infty)} t^{p-1}\mu(\{x\in X: f(x)>t\}) d\mu_t$ for any natural $p\ge 1$
27705
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
69656
Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
16053
Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$
150305
$f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
144192
Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions
61133
Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
147705
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
26658
A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
54266
Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$
101267
Show: $\int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt$
42366
Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
89421
Does $\int_{1}^{\infty} g(x)\ dx$ imply $\lim\limits_{x\to \infty}g(x)=0$?
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
108560
Proof of a Gaussian Integral property
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
106401
Prove: $p-mq | f(m)$ where $m$ is any integer