Showing that a function $u_\epsilon\rightharpoonup0$ in the $L^2$ norm, as...
Let $u : \mathbb{R} \to [-1, 1]$ be defined by $u(y) = \cos(2\pi y)$.Set $u_\epsilon(x) = u\left(\frac{x}{\epsilon}\right)$ for $x \in (a,> b)$, where $a, b \in \mathbb{R}$ and $a < b$. Show that...
View ArticleMethod of proof of $x^y
Is it valid to prove this statement with this method assuming that If $x>1$$x^p<x^q$ when p<q and p and q are rationalStatement: $x^y<x^z$ if $1<x$,$y<z$; \Attempt:Take the dedekind...
View ArticleConcavity of function implies convex upper contour
Today I saw a theorem in class that stated the following:$f$ is concave $ \Rightarrow\{z \in \mathbb R^n : f(z) \ge c\}$ is convex.The proof is relatively straight forward and I understand. However, I...
View Articlea measurable mapping is still measurable after completion of measure space
if f is a measurable mapping from $(\Omega,\mathcal{F},\mu)$ to $(E,\mathcal{E},\nu)$, then f is a measurable mapping from $(\Omega,\overline{\mathcal{F}},\mu)$ to $(E,\overline{\mathcal{E}},\nu)$,...
View ArticleVerification of formula for positive and negative variation in Folland's
Let $F\in BV$ be real-valued, and denote by $T_F(x)$ the total variation function of $F$, $$T_F(x)=\sup\left\{\sum_1^n |F(x_j)-F(x_{j-1})|:n\in\mathbb N,-\infty<x_0<\cdots<x_n=x\right\}.$$...
View ArticleIs $ G(x)=\int^\infty_{-\infty}...
While I am dealing with the non-local PDE problem$$((-\Delta)^{s}+I)u=f\qquad\text{in }\mathbb R^n$$I came across the following integral when I tried to compute its Green's...
View ArticleEquivalent definitions for $O_p$
I know that $Y_n=O_p(X_n)$ is by definition,$$\lim_{C \rightarrow \infty}\limsup_{n \rightarrow \infty}P(|Y_n|/|X_n| > C)=0. $$ I want to define a space $A_n$ such that $Y_n\leq C_0|X_n|$ for some...
View ArticleIntegration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let...
View ArticleBounded, measurable and supported on a set of finite measure function
Suppose f is a bounded and measurable function on R and supported on a set of finite measure. Prove that for every $\epsilon \gt 0$ there exists a simple function $s$ such that $\int |f-s|dx$ $\lt...
View ArticleSeveral variations of Ahmed integrals:...
The following are four integrals resembling Ahmed integrals:$$\begin{aligned}&I_1=\int_{0}^{1} \frac{1}{\left ( 1+y^2 \right )\sqrt{2+y^2} } \arcsin\left ( \sqrt{\frac{2-y^2}{4} } \right )...
View ArticleProve that if a function is increasing on [a,b] and satisfied the...
Let $f$ be increasing on $[a,b]$, i.e. for all $x<y$ in $[a,b]$, $f(x)\leq f(y)$. Also assume $f$ satisfies the intermediate value property. Show that $f$ is continuous on $[a,b]$.My attempt: Let...
View ArticleBounded almost-homomorphisms on the integers
Let $f : \mathbb{Z} \to \mathbb{Z}$ be an "almost-homomorphism": The set $\{f(n+m)-f(n)-f(m) : n,m \in \mathbb{Z}\}$ is bounded. We may assume that $f$ is odd, i.e. $f(-n)=-f(n)$ for all $n$. Assume...
View Article$\{(x,c)\in\mathbb{R}^n\times\mathbb{R}|f(x)>c\}$ is a subset of the interior...
I would like to ask if the following claim is correct?Claim$\quad$ Let $f:\mathbb{R}^n\to\mathbb{R}$ be a continuous function, then the interior of its hypograph...
View ArticleLocal surjectivity theorem
Let $f:\Omega \to \mathrm{R^n}$ be a continuous map, differentiable at $x_0$ and such that $Df_{x_0}$ is invertible. I want to prove there exists $\epsilon>0$: $f$ is surjective in...
View ArticleHow to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
I want to prove the following theorem (no idea whether it has a name):Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all $x...
View ArticleProof that a particular quantity is not the limit of a function.
I want to use the notation ε-δ and the definition of the limit of a function according to Cauchy to strictly show that if some quantity depends on the same variable as the function under study, then it...
View ArticleDecomposition $J$-measurable
We say that a J-decomposition $\mathcal{A} = \{A_r\}_{r=1}^s$ of a J-measurable set $C$ refines a J-decomposition $\mathcal{C} = \{C_i\}_{i=1}^k$ when for every $r \in \{1, \cdots, s\}$, there exists...
View ArticleA question about $\{\sup f_n > \alpha\}=\bigcup_{n=1}^\infty \{f_n> \alpha\}$
Let $\{f_n\}$ be a sequence of measurable function defines on a set $X$. it's simple to prove that $$\{\sup f_n > \alpha\}=\bigcup_{n=1}^\infty \{f_n> \alpha\},\tag1$$ where $\alpha\in...
View ArticleA series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$
Let $H_{n}$ be the nth harmonic number defined by $ H_{n} := \sum_{k=1}^{n} \frac{1}{k}$.How would you prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}?$$Simply replacing $H_{n}$ with...
View ArticleCompact support vs zero outside bounded set
In Advanced Calculus by Loomis and Sternberg, regarding functions $f : \mathbb{R}^{n} \rightarrow \mathbb{R}$, they frequently employ the condition that $f$“has compact support”, where the support of...
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