If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x)...
Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b]...
View ArticleIntegral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:$\bullet$$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a....
View ArticleA lemma of Cielsielski
Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901...
View Articlea limit for integrable functions
"If $f$ is integrable on $[0,A]$ for every $A>0$, and $\lim_{x\to\infty}f(x)=1$, then $\lim_{t\to 0^+} t\int_{0}^{\infty} e^{-tx}f(x)\: dx$ exists".(I'm convinced it's true, some examples suggest...
View ArticleSign permanence of locally Lipschitz functions calculated on a sequence
Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I...
View ArticleFunction property on balls and scaling
Assume $X$ is a compact metric space and $Y$ is a metric space. Assume $F:X\rightarrow Y$ is a map.Fix $L\geq 1$. Suppose for each $x\in X$, there exists an $\epsilon_x>0$ such that for every...
View ArticleCan anyone explain the constant in this relation
When the following function is plotted in desmos,$$y^x = \frac{(xy)^y}{k},$$Graph where $k = 1$, i.e. $y^x = (xy)^y$.As k approaches the number $\approx 2.89473713041139$ ($1/k \approx...
View ArticleIs composition of Lebesgue measurable function and addition Lebesgue measurable?
$f: \mathbb R^n\to \mathbb R^m$ Lebesgue measurable, is it true that $(x,y)\mapsto f(x+y)$ is Lebesgue measurable on $\mathbb R^{2n}$?I know in general $f \circ g$ is not Lebesgue measuable provided...
View ArticleLegendre transformation is a continuous map
I was reading about an article about the Legendre transformation in convex analysis. It was defined like this :Let $\varphi$ be a continuous function on $[0,+\infty]$ and convex on...
View ArticleDenseness of $C^{\infty}(M)$ within $C^k(M)$
Let $ M $ be a compact smooth manifold. For $ k \geq 1 $, let $ C^k(M) $ denote the space of real-valued functions on $ M $ of class $ C^k $, equipped with the uniform $ C^k $ norm—that is, the sum of...
View ArticleCauchy-Schwarz application: $\left(\int_1^ef(x)dx \right)^2 \leq \int_1^e...
Let $f:(0,\infty) \rightarrow \mathbb{R}$ be continuous. I need to show that $$\left(\int_1^ef(x)dx \right)^2 \leq \int_1^e xf(x)^2dx$$ I have been trying to use C-S to prove this but with no luck.
View ArticleProve $|f(x)-f(a)-df(a)(x-a)|\le \frac{M}{2}\|x-a\|^2$ when $\|d^2f(x)\|$ is...
Suppose $a\in \mathbb{R}^p$ and $f$ is a real-valued function whose second-order partial derivatives all exist and are continuous on $B_r(a)$. Also, suppose that the operator norm $\|d^2f(x)\|$ of the...
View ArticleHow to construct a sequence $\{g_n\}$ of $\mathscr{A}$-measurable simple...
I am in the middle of proving a result. It would be unnecessary to type the whole thing out. I just want to ask one step where I got stuck on.So let $(X,\mathscr{A},\mu)$ be a measure space. Suppose...
View ArticleTwo equivalent definitions of almost sure convergence of random variables.
Let $\{X_n\}_{n=1,2,\cdots}$ is a sequence of random variables. There are equivalent definitions of almost sure convergence of random variables. How can one prove the...
View ArticleAlmost sure convergence and equivalent definition
First of all this question has related answers here and here, but I am still struggling to understand the proof and the nuances that go along with it. Show that a.s convergence of a sequence of random...
View ArticleProve of rules of operation with limits that diverge
Could someone help to check my proof, I'm not sure if they are rigorous:Given ${a_n}$, ${b_n}$, and ${c_n}$ be sequences of real numbers and let k be a constant, $\lim_{n \to \infty} a_n = \infty$,...
View ArticleI don't know how to prove this limit
Let $x=(x_1,\cdots,x_n)$, $\alpha=(\alpha_1,\cdots,\alpha_n)$$\sum_{i=1}^n\alpha_i=1$, we have $\alpha>0,x_i\ge 0$ for all $i$. Define $$M_t(x,\alpha)=\left(\sum_{i=1}^n\alpha_ix_i^t\right)^\frac...
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Doubts about some problems concerning comparability of infinitesimals?
I'm reading Efimov's Higher Mathematics. There is this definition:After that, there are several problems with some proofs, computational exercises, etc. And then, there are the following problems:I'm a...
View ArticleAnalysis research project ideas [closed]
Before I ask my question, this is my first post on MSE, so I apologise if I have not met the standard etiquette of a post.I have finished my first year at a UK univeristy for a maths degree. In our...
View Article$u\in H^1(\mathbb{R}^N)\cap C^1 \Rightarrow u$ vanishes at infinity?
Suppose $u\in H^1(\mathbb{R}^N)\cap C^1 $, where $H^1$ is the Hilbert-Sobolev space. Is it true that the limit of $u$ at infinity in any direction is 0?
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