# Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

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### $f_n=\frac{1}{x^{\frac{1}{n}}}$ uniformly converges to 1 [closed]

This is the problem that I am trying to prove. Show that $f_n(x)=\frac{1}{x^{\frac{1}{n}}}$ where $x\in (0,\frac{1}{2})$ is uniformly convergent to $f(x)=1$. It will be great if someone can tell me ...
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### Almost everywhere convergence in product measure and that in coordinate ones

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
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### CDF of a convergent positive series

Let $Y_0, Y_1, \ldots$ be an i.i.d. random sequence such that $$\mathbb{P}(Y_k = 0) \;=\; 1 - \mathbb{P}(Y_k = 1) \;=\; p \qquad \text{for each k\ge 0}.$$ I am interested in the following random ...
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### Pointwise convergence of a function where convergence to multiple values occur at a single point

Consider, $$f_n(x)=\frac{1-nx^2}{(1+nx^2)^2}$$ where, $$x \in \mathbb{R}, n \in \mathbb{N}$$ It is clear to me that not including $x = 0$, each function point converges to $0$ as $n \to \infty$. The ...
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Consider a measurable space $\big( E \times F, \mathcal{E} \otimes \mathcal{F} \big)$, an $(\mathcal{E} \otimes \mathcal{F})$-measurable and positive function $f$, and a measure $\nu$ on $\big( F, \... 2 votes 2 answers 54 views ### Uniform convergence of$\sin(nx) \cdot\frac{x}{n} + x$I've shown that$f_n(x) = \sin(nx)\cdot \frac{x}{n} + x$converges pointwise to$f(x) = x$. Now if I consider$|f_n(x) - f(x)| = |\frac{x}{n}\sin(nx)|$it looks that it is dependent over$x$so I ... 0 votes 0 answers 15 views ### Applying convergence space to "convergence in measure" and processes I read up on "Convergence space" on Wikipedia and An initiation into convergence theory - Szymon Dolecki. I think I get the general idea. However, I study within applied math, so I am trying ... 2 votes 1 answer 75 views ### Pointwise convergence of generalized inverse function [duplicate] I am reading Resnick's Extreme Values, Regular Variation, and Point Processes. In chapter 0.2 he writes about the generalized inverse of a non-decrasing function F: $$F^{\leftarrow}(y):=\inf\{x:F(x)\... 0 votes 1 answer 19 views ### Bound for the difference between two functions Consider f_1, f_2, g_1, g_2 four continuous functions defined on the real line. I know that for every x \in \mathbb{R}$$0 \leq |f_1(x)| \leq |g_1(x)|, \quad 0 \leq |f_2(x)| \leq |g_2(x)|$$If I ... 0 votes 1 answer 35 views ### If a sequence of finite signed measures converges the sequence of their "maximizers" converges? F^1, F^2 are the CDFs corresponding to two distributions corresponding to positive finite measures, supported in [0,1], i.e. F^i(x) is the measure of the set [0,x] under the i-th measure. ... 2 votes 0 answers 45 views ### Weak Convergence implies Pointwise Convergence (on a Countable set) Let \mathcal{C} = \{x_1, x_2, \cdots\} be a countable set of \mathbb{R}. Let \{\mathbb{P}_n\} and \mathbb{P} be probability measures on \mathcal{C}. Prove that \mathbb{P}_n \stackrel{w}{\... 4 votes 2 answers 68 views ### Probability convergence of a martingale defined as iid random variable product Let (\beta_n)_{n \geq 1} be positive independent and identically distributed random variables with \mathbb{E}[\beta_1] = 1 and \mathbb{P}[\beta_1 < 1 ] > 0. Define the martingale M_n = \... 1 vote 0 answers 90 views ### Fundamental theorem of \Gamma-convergence In the paper "A handbook of Γ-convergence" I've read the following: "This is the fundamental theorem of Γ-convergence, that is summarized by the implication Γ-convergence + ... 1 vote 1 answer 32 views ### Pointwise and uniform convergence for particularly well-behaved functions I'm asking myself... if f_n(x):\mathbb{R}\to\mathbb{R} are infinitely differentiable functions, and each f_n is such that f_n(x)\underbrace{=}_{|x|\to\infty}\mathcal{O}(|x|^{-N}) for any N\in\... 0 votes 0 answers 24 views ### prove there is no pointwise convergence of \sin(nx) [duplicate] I was reading a calculus book and there was this task: prove that you cannot select a pointwise convergent subsequence from the sequence$$f_n(x) = \sin(nx)$$consider that it's on$$[0, 1]$$I've ... 2 votes 1 answer 116 views ### Initial Topology and Weak and Strong Operator Topology Given a set$X$and a family of topological spaces$(Y_i)_{i \in I}$and functions$f_i : X \to Y_i$, we can define a topology on$X$called the initial topology on$X$with respect to the topological ... 0 votes 0 answers 30 views ### Convergence almost everywhere criterion I was reading Loeve's Probability Theory and I saw the following criterion for determining convergence almost everywhere: Convergence a.e. criterion. Let$X, X_n$be finite measurable functions.$ ...
Given $a,b>0$, let $f_n:\mathbb{R}\to\mathbb{R}$ for $n>0$ be the random variable $t\mapsto(1/\sqrt{n})\sum_{k<n}\sin(t/(a+kb/n)+\phi_k)$ where $\phi$ is uniformly distributed in $[0,2\pi)^n$....