Relation between $(\Omega, \sigma(X, Y), \mathbb{P})$ and $(\Omega \times...
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be probability space, $(S_1, \mathcal{S_1}), (S_2, \mathcal{S_2})$ be measureable spaces, and $X:\Omega \rightarrow S_1$, $Y:\Omega \rightarrow S_2$ be...
View ArticleRatio test for sequence convergence
The convergence part of the ratio test for real sequences says, roughly, that if the absolute values of the ratios of a given (non-zero) sequence $a_n$ are, for sufficiently large $n$, bounded below 1,...
View ArticleRational number representation in quotient of shifted primes [duplicate]
Prove that every positive rational numbers r can be written as $r={(p+1)}/{(q+1)}$ where p and q both prime number. I find this problem in the long form math analysis book by J Cummins in the open...
View ArticleLimit of ratio of two equivalent sequences
I was working through a proof of the Cartan's closed subgroup theorem, from Lie Theory.I got stuck in a passage when sequences of real numbers are used.I don't think I need to give you all of the...
View ArticleApplication of Implicit Function Theorem
So I'm a set of practice problems regarding this but I don't quite understand how to think about this...Example of a problem: $x^3 (y^3 +z^3 )=0$ and $(x-y)^3 -z^2 -7=0$A point lies on the surfaces...
View ArticleNeither the toughest, nor the easiest integral $ \int_0^1 \frac{\log...
Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegantapproach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$$$=4...
View Articleif $\int g\phi =\int f\phi$ for all $\phi \in C^{\infty}_{c}(U)$, then $f=g...
In Folland's real analysis textbook, on page 283, he mentioned that for an open set $U \subset \mathbb{R}^{n}$, and $f,g \in L_{loc}^{1}(U)$ .$f$ and $g$ define the same distribution precisely when...
View ArticleThe difference between Riemann integrable function and Lebesgue integrable...
My professor asked me how to intuitively understand Lebesgue's dominated convergence theorem and what's the effect of the integrable dominated function. More specifically, when we are given a Lebesgue...
View ArticleIntegrability of Thomae's Function on $[0,1]$.
Consider the function $f: [0,1] \to \mathbb{R}$ where$f(x)=$\begin{cases}\frac 1q & \text{if } x\in \mathbb{Q} \text{ and } x=\frac pq \text{ in lowest terms}\\0 &...
View ArticleOrder of proofs for limits, derivatives, and series of e and $ln x$
In high school calculus, the focus is on computations and not proofs. Proofs are introduced (if at all) for the sake of understanding rather than rigour.As this answer mentions, proofs at this level...
View ArticleMeasurable of a vector-valued function
I am reading the Book of Stein,Singular integrals.In page 45:Let $\mathcal{H}$ be a separable Hilbert space.Then a function $f(x)$, from $\mathbb{R}^n$ to $\mathcal{H}$ is measurableif the scalar...
View ArticleIf a series $\sum_{n=1}^{\infty} a_n$ is Cesàro summable &...
If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges gives the proof of the easier version of the proposition, but I have no clue...
View ArticleAsymptotic behavior of the Hermite functions
I would like to understand the asymptotic behavior of the Hermite function :$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$where $H_k(x)$ is the $k-$th Hermite polynomial. For...
View ArticleHow to calculate a lateral limits for $\mathbb{f}(x) = \frac{1}{x^2}...
How should I go about calculating the lateral limits in $x_0 = 0$ of$\mathcal{f}(x) = \frac{1}{x^2} \sin\left(\frac{1}{x}\right)$, so I have to calculate:$\lim_{x \to 0^+} \frac{1}{x^2}...
View ArticleUniform Tightness of a Sequence of Probability Measures in $\mathbb...
Consider a sequence $(P_n)_{n\in\mathbb N}$ of probability measures on $(\mathbb R^{\mathbb N}, \mathcal B)$, where $\mathbb R^{\mathbb N}$ is the countable Cartesian product of $\mathbb R$, and...
View Articleestimate of the cutoff function in the proof of inner regularity of Poisson...
In the proof of theorem2 in: https://www.math.ualberta.ca/~xinweiyu/527.1.11f/lec14.pdf , I am struggling with the estimate of the cutoff function.Theorem: Let $u \in W^{1,2}(\Omega) $ be a weak...
View ArticleCutoff function in the proof of inner regularity of Poisson equation
In chapter 11 of Jost partial differential equations he wants to prove the following theorem of interior regularity for the Poisson equation:Theorem 11.2.1: Let $u\in W^{1,2}(\Omega)$ be a weak...
View ArticleLimit with sequence. How to "try them all"?
I don't understand how to operate when I have to deal with limits of function by using sequences. We have studies this theorem that says that $\lim_{x\to x_0} f(x)$ exists if by choosing any sequence...
View Articleques of Riemann integration [closed]
Consider the characteristic function of a single-point set, say l = X{s},defined by l(x) = { 1 if x =5' } · Prove that l is integrable on [2, 9]0 otherwiseand find l; f
View ArticleTwice differentiable at only one point?
We know the famous $x\mapsto x^2 1_{\mathbb Q}(x)$ which is differentiable at just one point. But is it possible for a function to be twice differentiable only at one point, say $x = c$? For this, I...
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