$\mathbb{R}\setminus\mathbb{Q}$ as a union of $F_\sigma$ set and a set of...
Every measurable set in a locally compact, $\sigma$-compact, Hausdorff space $X$, under a suitably nice regular measure, is the union of an $F_\sigma$ and a set of measure zero; this is well-known....
View ArticleDerivative of $x^TA$
Let be $A$ an $n\times n$ matrix and $x\in\mathbb{R}^n$. Then wikipedia and the books say that the derivative of $f(x):=x^TA$ is $A^T$.I was trying to apply this result to the official definition of...
View ArticleIs their any interconnection between the Jordan measure and Hausdorff measure?
In short, while Jordan measure is useful for regular sets with well-behaved boundaries, it fails for fractals, and the Hausdorff measure is the correct tool to use in these cases.I know that Jordan...
View ArticleHilbert space question: if $M=\{f\in L^2(0,+\infty)\ |\ \int_0^\infty...
This was from a Real Analysis exam:In the Hilbert space $H=L^2(0,+\infty)$. Consider$$M=\left\{f\in L^2(0,+\infty)\ \bigg|\ \int_0^\infty f^2(x)e^xdx<+\infty\right\}$$Find $M^\perp$.Since the...
View ArticleWhy is $\ell^\infty(\mathbb{N})$ not separable?
My functional analysis textbook says "The metric space $l^\infty$ is not separable."The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is...
View ArticleConditions for the existence of the second Derivative
I was given the function $|x| - 1$ on the interval $[-2,2]$ and asked to integrate it.My answer was the following piece-wise function:$y = -\frac{1}{2}x^2 - x - \frac{1}{2},$ for $-2 \le x < 0$$y =...
View ArticleEquivalent formulations of strict positivity of a symmetric operator
Suppose $\mathcal{H}$ is a separable Hilbert space, and $A: \mathcal{H} \mapsto \mathcal{H}$ is a symmetric linear operator. We say $A$ is a strictly positive definite operator if $\langle Ax, x...
View Articlecomputing $\frac{1}{2\pi...
I was reading this and got stuck right before Remark $1$.Let$$F(t)=\frac{1}{2\pi i}\int\limits_{(1)}\frac{e^{(1-t)s}}{s}\Psi\left(\frac{\Im (s)}{\log x}\right)ds$$where $\Psi$ is a smooth cut off...
View ArticleLimes of series of polygons with $\leq M$ edges also has $\leq M$ edges
Suppose in the following, all polygons are $C^0([0,1],\mathbb{R}^2)$. If the series of polygons $(P_n)_{n\geq 1}$ with all having $\leq M$ edges for $M\geq 1$ is (uniformly) convergent to the limes...
View ArticleConvergence of Lipschitz constant $\big[h(T^n)\big]^{\frac{1}{n}}$
In this context, we will use the symbol $h(T)$ to denote the Lipschitz constant of a Lipschitz map $T$.Let $X$ be a completed metric space and $T:\ X\longrightarrow X$ is a Lipschitz map. Then the...
View ArticleWhy do Dedekind cuts have no maximum?
There is a property of Dedekind cuts that states that there exists no maximum $a_0\in A$ for all cuts $A\in\mathbb{R}$.My question is: Why do Dedekind cuts need this property?What would happen if...
View ArticleLocal Boundedness and Lebesgue Integrability [closed]
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function. Suppose that for every $\epsilon > 0$, there exists a compact set $K \subset \mathbb{R}$ such that $m(f^{-1}((-\infty,...
View ArticleHow can I prove the existence of multiplicative inverses for the complex...
If the complex number system is defined to be the set C = R x R (called the complex numbers) together with the two functions from C x C into C, denoted by + and * (multiplication), that are given by...
View ArticleInterpretation of a sigma-algebra not generated by a random variable.
I am wondering what is the interpretation of a sigma-algebra that is not generated by a random variable?For example if we have a random variable $X$ on a probability space $(\Omega, \mathcal{A},P)$...
View ArticleEtymology of the term "sigma-finite" measure
As many people on this site know, and could probably recite by heart at any minute, the definition of a $\sigma$-finite measure is as follows:Let $(X, \mathcal{F}, \mu)$ be a measure space. The measure...
View ArticleProve that finite intersection of open sets is open using open rectangles
Consider a finite collection of open subsets of $\mathbb{R}^n,$$\{S_i\}_{i=1}^{m}$. I want to prove that $I:=\bigcap_{i=1}^mS_i$ is open.I know how to prove this statement using open balls. You take an...
View ArticleDoes $\alpha\leq \beta+ \gamma$ imply $\sin \alpha \leq \sin \beta+\sin\gamma$?
Here is the question: Suppose that $\alpha,\beta,\gamma\in [0,\frac{\pi}{2}]$ and $\alpha\leq \beta + \gamma$. Is it true that $\sin \alpha \leq \sin \beta+\sin\gamma$? I do not really know how to...
View ArticleHardy's inequality holds for index $\alpha
Suppose $f(0)=0$ and $f\in H^1(\mathbb{R}_+,xdx)$, i.e.,$$ \left\|f\right\|_{H^1(\mathbb{R}_+,xdx)}:=\left\|x^{1/2}f\right\|_{L^2(\mathbb{R}_+)} +...
View ArticleAn integral inequality (one variable)
Anyone has an idea to prove the following inequality?Let $g:\left(0,1\right)\rightarrow\mathbb{R}$ be twice differentiableand $r\in\left(0,1\right)$ such...
View ArticleExpectation of maximum of $n$ Pareto random variables
Let $X$ be a Pareto random variable with scale parameter $x_m > 0$ and shape parameter $\alpha > 1$, i.e., $\mathbb{P}(X > x) = (x_m / x)^\alpha$. Let $X_1,X_2, \dots, X_n$ be i.i.d copies of...
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