Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.
9159
$L^p$-norm of a non-negative measurable function
98098
Show that $\mu_{F}$ is a measure on $(\mathbb{R},\mathcal{S})$.
43956
Differentiating $f(x)^{g(x)}$
63309
How to find a measurable but not integrable function or a positive integrable function?
91126
A question about Measurable function
40564
Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
119735
Show that $\mathbb{E}(g(T-p)) < \mathbb{E}(g(S-p))$ for any convex function $g$ if $T$ and $S$ are estimators of $p$
109301
Lebesgue Integral of Non-Measurable Function
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
131793
Zero integral of measurable $f$ on every interval implies $f=0$?
150968
Differentiable function has measurable derivative?
109570
How to show that the set of points of continuity is a $G_{\delta}$
157485
Integral of limit of pointwise convergent sequence of non-negative measurable functions
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
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
103513
Show that $\lim_{n\to \infty}\int_{-\pi}^{\pi}f(x)g(nx)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\int_{-\pi}^{\pi}g(x)\,dx$
123806
How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?