query: how to prove this equality $||f||_{L^p}^{p}=p\int_0^{+\infty} \lambda^{p -1}\mu(E^f_\lambda) d\lambda$
positive_passages.docid: 27705
positive_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 38761
negative_passages.text: Is the $ L^{p}$$[0,1]$ norm continuous in p?
negative_passages.docid: 5350
negative_passages.text: Explain why $E(|X|^p)=p\int_{0}^{\infty}y^{p-1}P(|X|>y) dy$
negative_passages.docid: 39797
negative_passages.text: Convergence of integrals in $L^p$
negative_passages.docid: 47895
negative_passages.text: Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 110275
negative_passages.text: Let $p>0$. Show the $\lim_{n\to\infty}(a^n/n^p)$ is equal to:
negative_passages.docid: 94119
negative_passages.text: Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
negative_passages.docid: 98710
negative_passages.text: Finding $\int_0^{\infty}xe^{-\lambda x} \, dx$
negative_passages.docid: 142785
negative_passages.text: How can I prove Holder inequality for $0<p<1$?
negative_passages.docid: 61162
negative_passages.text: Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
negative_passages.docid: 40564
negative_passages.text: Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
negative_passages.docid: 40936
negative_passages.text: How prove this integral inequality
negative_passages.docid: 151400
negative_passages.text: Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 62192
negative_passages.text: Proof $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty {X(u)du} $
negative_passages.docid: 1588
negative_passages.text: How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$