To show $\lim_{x \to \infty} \int_{x}^{x+1}f(t)dt=0$ if $\lim_{x\to\infty}...
$f:\mathbb{R}\to \mathbb{R}$ is Riemann integrable on any bounded interval and $\lim_{x\to\infty} f(x)=0$.Define $g(x)=\int_{x}^{x+1}f(t)dt$, we need to show $\lim_{x\to\infty} g(x)=0$.Please give me...
View ArticleThe measurability of convex sets
How to prove the measurability of convex sets in $R^n$? I have seen a proof, but too long and not very intuitive. If you have seen any, please post it here.
View ArticleIs there a measure that produces given values (probabilities or cardinals)...
Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k,...
View ArticleFinding a non-affine function satisfying symmetry properties
I am looking for an example of a continuous, non-affine function $u\colon X\to \mathbb{R}$ and a continuous, non-negative function $\epsilon\colon X\to\mathbb{R}_{\geq 0}$ such that the following hold...
View ArticleAlternative proof $g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$...
The given inequality$$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$, comes out from an answer given to this other recent question.The...
View ArticlePulling out a new variable inside an integral
Suppose I have$$f(t)\le\int_0^t\frac{(f(s)+c)}{g(s)}ds$$ where $c$ is constant.Then, by writing $f(t)=\int_0^tf'(s)ds=\int_0^tdf(s)$$=\int_0^td(f(s)+c)$, is it possible to write the above inequality...
View Article$\sigma$-finite measure $L^p$ space is isometric to a finite measure $L^p$ space
I have a measure space $(X,M,\mu)$ where $\mu$ is $\sigma$-finite. Then there exist a finite measure $\lambda$ s.t. the space $L^p(\mu)$ is isometric to $L^p(\lambda)$.First I proved that there exist a...
View ArticleOscillations of Lagrange interpolation polynomials
Let $I = [a,b]$ be a real closed interval. Let $n$ be a positive integer and let $x_i = a+i\frac{(b-a)}{n}$ for $i=0,...,n$. Let $p_j(x)$ be the Lagrange interpolating polynomial of the $n+1$ points of...
View ArticleInitial and boundary conditions for parabolic PDEs
Consider a parabolic PDE of the form\begin{align}\phi_t=f\left(\phi,\phi_x,\phi_{xx} \right),\end{align}with $f$ some reasonable function, and if needed for the argument below linear $\phi_{xx}$.It is...
View ArticleExamples of continuous functions that are monotone along all lines
I am looking for different examples (or even a complete characterization if this is possible) of continuous functions that are monotone along all lines. By that I mean functions $f\colon...
View ArticleConvolution of mixture of Erlangs
Let $P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax}$ be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$. Is is possible to determine an analytic expression for...
View ArticleAnalysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where...
Context. Let $C^k(\mathbb R^n)$ denote the space of functions defined on $\mathbb R^n$ that are $k$ times continuously differentiable, where $k \geqslant 1$ is an integer. As usual, define...
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Proof in measure theory
I have found excellent proof about changing variables in polar coordinateThis is the problemLet $S^{n-1}=\left\{x \in R^2:|x|=1\right\}$ and for any Borel set $E \in S^{n-1}$ set $E *=\{r \theta:...
View ArticleConvergence to zero in L2 implies probability of being in an open set also...
Suppose I have a sequence of random variables $\{X_n\}_{n=1}^{\infty}$ taking values in $\mathbb{R}^{k\times k}$ and $S \subset \mathbb{R}^{k\times k}$ is a given open set. If I know...
View Articleupper bound for $\int \frac{1}{|x-y|}$
Let $\alpha>0$, determine some upper bound for this integral for $x\in B_{1}$, $$\int_{B_{1}\cap B_{(1+1/\alpha)|x|}}\frac{1}{|x-y|}\,dy.$$My approach: If $x\in B_{1}$ then $|x|\leq 1$ and...
View ArticleMaximum point of a function about binomial coefficients
For a fixed positive integer $m$, define $$g(d)=\sum_{j=d+1}^m\binom{m}{j}\binom{j-1}{d},\ \ 1\leq d\leq m-1.$$I used MATLAB to calculate the value of $g$ and it seems that $g(d)$ is maximal iff...
View ArticleProof: If $f\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$, then $\{x\in...
I am self-studying Measure Theory by Donald Cohn. When proving this statement:If $f\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$, then $\{x\in X:|f(x)|>\|f\|_{\infty}\}$ is locally $\mu$-null.the book...
View ArticleImproper multiple integral of vector valued function and corresponding...
Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be an appropriate function.I want to know the definition of the improper Riemann integral\begin{equation}\int_{\mathbb{R}^n}f,\end{equation}and when we can...
View ArticleIs the set $\{ x\in \mathbb{Q}: 2< x^2
Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$ ?I think $\{ x\in \mathbb{Q}: 2< x^2 <3\}=\{ x\in \mathbb{Q}: 2\leq x^2 \leq 3\}$, so it is bounded and...
View ArticleProve that $\delta_{ij}\delta_{jk}=\delta_{ik}$
Firstly, I should mention that I have just started learning about tensors, the problem is that I need to understand why the result $$\fbox{$\delta_{ij}\delta_{jk}=\delta_{ik}$}\tag{1}$$ is true in...
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