# Tagged Questions

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### Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
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### If the integral $\int_0^\infty xf(x)\,dx$ converges, so does $\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx$

Let be $f:[0, \infty)\rightarrow [0,\infty)$ a measurable function such that $$\int_0^{\infty}x\cdot f(x)\,dx< \infty.$$ Show that $$\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx<\infty .$$ ...
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### A Fubini-Tonelli's Theorem Problem

Let $E \subseteq \mathbb{R^n}$ a measurable set, such as for almost every $x \in \mathbb{R^n}$ we have $|E \triangle (E+x)|=0$ (Where $\triangle$ means simetric difference between two sets and ...
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### Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Let $(X,A,μ)$ be a finite measure space. Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$ a.) Prove that if $f_n → 0$ in measure, then $f_n → 0$ in $L_1(μ).$ b.) If ...
### Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$
As part of an analysis qual problem, I am having a hard time showing that $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$. Any suggestions? Thanks in advance. I ...