Questions tagged [sequence-of-function]
Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.
985
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Is the product of two uniformly convergent sequence of functions uniformly convergent?
I am doing the exercise V.1.7 of Amann and Escher's Analysis. In the exercise I am required to prove that the product of two uniformly convergent sequence of functions is uniformly convergent if just ...
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2
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51
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Counterexample for a convergent sequence of functions on a $\sigma$-finite measure space
I have already proven following statement, but I am struggling to construct a counterexample, even with the hint. Every kind of help is appreciated:
Let $(X, A, \mu)$ be a measure space with $\mu(X) &...
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A question for uniform convergent of sequence of functions.
Consider the sequence of functions $<f_n(t)>$, defined as
$
f_n(t) =
\begin{cases}
e^{-t^2} & \text{if } -n \leq t \leq n \\
\frac{e^{-n^2}}{[1-n(t-n)]} & \text{if } n \leq t &...
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23
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Generalization of Leibniz criterion of uniform convergence for sequence of functions [closed]
I want to know the proof of this "generalization" version of Leibniz criterion of convergence.
Let $\{f_n(x)\}_{n\geq 1}$ be the sequence of functions on $A\subset\mathbb{R}$ such that:
(i) $...
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Questions and observations regarding Putnam 2020 - A.6.
CONTEXT
My starting point is Question A.6 of Putnam 2020 competition, that goes like that
For a positive integer $N$, let $f_N$ be the function defined by
$$
f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)...
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Show that $B_1(0)$ is a closed set in the space $C([0,1])$
Let be $(C([0,1]),\Vert \cdot\Vert_{\infty})$ the normed space of continuous functions, equipped with the supremum norm $\Vert\cdot\Vert=\sup\limits_{x\in[0,1]}|f(x)|$.
Show that $B_1(0):=\{f\in C([0,...
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71
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Does the sum of a sequence of functions $\{f_n(x)\}$ converge if the upper bound $h_n(x)$ with $f_n(x)\le h_n(x)$ converges pointwise to $0$?
I am dealing with a problem on the existence of the limit for a sequence of function $\{g_n(x),n\in\mathbb N\}$ with function $g_n(x):\mathbb R^n\to\mathbb R^m$, i.e. to judge the existence of limit $...
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Does the existence of limit of a sequence formed by continuous functions at some points imply the existence of the limit at other points?
$\{A_i,i\in\mathbb N\}$ is a fixed matrix sequence with element $A_i\in \mathbb R^{n\times m}$.
$\Phi\in\mathbb R^{m\times m}$ is a constant matrix and $d\in\mathbb R^m$ is a vector.
The sequence $\{...
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1
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What is the pattern so we can make the next stars?
I found the following pattern question in a group! It took me a lot of time but unfortunately I don't have any ideas to find any logical thing here.
Here's the picture of the question:
The question ...
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42
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$f(x)=x^2-x+\frac{1}{6}$ on $[0,1]$ and extend to $\mathbb R$ periodically. Prove that $f(x)=\frac{1}{2\pi^2}\sum_{n\neq 0}\frac{1}{n^2}e^{2\pi inx}$.
Let $f(x)=x^2-x+\frac{1}{6}$ on $[0,1]$ and extend to $\mathbb R$ periodically.
(a) Prove that $f(x)=\frac{1}{2\pi^2}\sum_{n\neq 0}\frac{1}{n^2}e^{2\pi inx}$.
(b) Prove that for any integer $M\geq 1$, ...
- 307
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1
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Find a sequence of functions
There are familiar functions whose derivatives are periodic. $e^x$ has a period of $1$, since
$$\frac{d}{dx} e^x = e^x$$
And $\sin(x)$ has a period of $4$. I am interested in finding a sequence of ...
- 2,038
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Uniform convergence of sequence of functions gn and fn
Given two functions
$$f_n (x)= x\left(1+\frac{1}{n} \right), x \in \mathbb{R}, n \in \mathbb{N}$$
And
$$g_n (x)= \begin{cases} \frac{1}{n} ,& x \space \text{is irrational} \space \text{or} \space ...
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Is $\left\{\sqrt{n}\sin^{\circ n}\left(\dfrac{z}{\sqrt{n}}\right)\right\}$ locally uniformly convergent over $\mathbb{C}-\{y{\rm i}:|y|\ge\sqrt{3}\}$?
Note $\sin^{\circ n}x$ as the $n$-fold iteration of $\sin x$. It is shown in this great answer that
$$
\lim_{n\to\infty}\sqrt{n}\sin^{\circ n}\left(\dfrac{x}{\sqrt{n}}\right)=\dfrac{\sqrt{3}x}{\sqrt{x^...
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29
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Term by term integration and uniform convergence
I have three queries:
(1) Do term by term integration always imply the order of integral-summation, integral-limit are interchangable and vice-versa?
(2) Suppose $\{f_n\}$ be a boundedly convergent ...
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How to prove that $ \lim_{n \to +\infty} \frac{x^n}{n!} = 0$? [duplicate]
I was doing some practice problems for limits and I encountered the following problem:
Does $\displaystyle\lim_{n \to +\infty} \frac{x^n}{n!} = 0$ exist, if so find the limit as a function of $x$.
I ...
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16
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generating sequences of functions
I have read the following theorem from Apostol's analysis.
Let $\alpha$ be of bounded variation on $[a, b]$. Assume that each term of the sequence $\{f_n\}$ is real valued function such that $f_n \in ...
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Prove or disprove that if $g$ is continuous, then $\lim_{n\to \infty}\int_{0}^{1}f_{n}(x)g(x)dx=0$
Here's the complete question:
Define a squence of functions $\{f_{n}(x)\}$ on $[0,1]$ as:
$$f_{n}(x)=\left\{ \begin{array}[lll] 11 & \mbox{if} & x=0\\ 1 & \mbox{if}& x\in (\frac{2k}{2^{...
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Is there a sequence of $C^2$ functions approximating $\max(0,x)$?
I know that the sequence
\begin{equation*}
f_n(x):=\begin{cases}\sqrt{x^2+\frac{1}{n^2}}-\frac{1}{n},& x\ge0, \\\\ 0,&\text{otherwise},\end{cases}
\end{equation*}
is a $C^1$ sequence pointwise ...
0
votes
1
answer
82
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Zeros in a sequence
Define the sequence $x_n$ by $x_1=0$ and
$$
x_n=x_{\lfloor n / 2\rfloor}+(-1)^{n(n+1) / 2}
$$
for $n \geq 2$. Find the number of $n \leq 2023$ such that $x_n=0$.
My approach was the following:
To find ...
- 1,053
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0
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6
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Sequence of sets of p-quantiles converging to a given set of p-quantiles
One of my homework problems asks us to show that a sequence of sets of p-quantiles converges to a given set of p-quantiles. I can start the question but don't know how to continue. Here's the problem ...
- 1
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1
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55
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Conditions for monotone convergence
I have a difference function of the form $ z_{t+1} = (1-\beta)z_t + \lambda(1-z_t) $ where $ 0 \leq z_0 \leq 1, 0 < \beta < 1 , 0 < \lambda < 1$. Find conditions such that the sequence ${...
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A doubt on uniform limit of a sequence of uniformly continuous functions
I was looking at this question is MSE.
Suppose $f_n:[0,1] \to \mathbb{R}$ is a sequence of uniformly continous functions and $f_n\to f$ uniformly.
These are the questions that I put myself.
What can ...
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If $f:[0,1]\to\mathbb R$ is continuous, then $f_n(x) = f(x^n)$ converges uniformly on $[0,a],$ $a < 1$ and $∫_0^1 f_n(x)\,dx \to f(0).$
Given f continues function in [0,1] we define a sequence of functions fn(x) = f(x^n) , given that for every number 0<a<1 the sequence of function is uniformly convergs in [0,a] to f(0) prove ...
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Construct a non-increasing sequence (An) such that it converges to its supremum. [closed]
We know that a real sequence which is increasing and bounded above converges to its supremum. Is this possible for a non - increasing sequence as well? State that sequence.
I am not able to think of a ...
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1
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Why is $C([0;1])$ with the supremum metric complete, but it is not with the integral metric
I am currently studying metric spaces in my mathematical analysis course and I came across two examples:
First - show that the set of continuous functions on a closed interval (denoted as $C([a;b]$) ...
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1
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Has the composition of a sequence of quasiconformal mappings with unbounded dilatation and another q.c. mapping still unbounded dilatation?
Let $G \subseteq \mathbb{C}$ be a bounded domain in $\mathbb{C}$. Consider for $m \in \mathbb{N}$ a sequence of quasiconformal mappings $f_m: G \longrightarrow \mathbb{C}$ with unbounded maximal ...
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Uniform convergence of the given series in the domain $\Bbb R$ [duplicate]
Check whether the following series of function is uniformly convergent in $\Bbb R$.
$$\sum_{n=1}^\infty (-1)^n \frac{x^2+n}{n^2}.$$
If the ddomain is any bounded interval then by Weierstrass M-test ...
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1
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Show that if a_ndenotes the $n$th positive integer that is not a perfect square, then $a_n = n+\{\sqrt{n}\}$
Show that if $a_n$denotes the $n$th positive integer that is not a perfect square, then $$a_n = n+\{\sqrt{n}\}$$ where {x} denotes the integer closest to the real number x.
$a_3=5=3+2$
$a_4=6=4+2$
$...
- 107
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2
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$\int _0^1 f(x)g_n (x) dx \rightarrow 0 $ as $n\rightarrow 0$
Question:
Prove $\int _0^1 f(x)g_n (x) dx \rightarrow 0 $ as $n\rightarrow 0$ for $f\in \mathcal{L}^1([0,1])$ and $\{g_n\}_{n\in\mathbb{N}}$ a sequence of measurable functions on $[0,1]$ such that
(i) ...
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0
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47
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Uniform convergence of a sequence of holomorphic functions on the unit circle and disc
A previous complex analysis qualifying exam problem:
Let $\{f_n\}_{n=1}^\infty$ be a sequence of functions holomorphic on the open unit disc $D = \{z:|z|<1\}$ and continuous on the closed unit disc ...
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$f_n$ is continuous on $[a,b]$, and for any $x\in [a,b]$, $\{f_n(x)\}$ is a bounded sequence. Show that for some interval, $f_n$ is uniformly bounded
$f_n$ is continuous on $[a,b]$, and for any $x\in [a,b]$, $\{f_n(x)\}$ is a bounded sequence. Show that for some interval, $f_n$ is uniformly bounded.
What to do? Heine-Borel property? How to find ...
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Rudin $7.14$ linked to $7.9$ why does Rudin make boundedness compulsory in uniform convergence?
In Rudin $7.14$, Rudin says
Theorem $7.9$ can be rephrased as follow: a sequence $\{f_n\}$ converge to $f$ with respect to the metric $\mathscr C(X)$ if and only if $f_n \rightarrow f$ uniformly on $...
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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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41
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Differentiability of the limits of sequences of functions between Banach spaces
$\newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}$Let $(E, |\cdot|_E)$ be a real Banach space and $X$ an ...
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Differentiability of the limits of sequences of functions from $\mathbb R$ to a Banach space
Let $(E, \|\cdot\|)$ be a real Banach space and $X$ an open subset of $\mathbb R$. I'm trying to prove below result which is used in the proof of Hille-Yosida theorem in Brezis' Functional Analysis.
...
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Can a convergent sequence have divergent subsequences? [duplicate]
I cannot seem to think of any example of a convergent sequence that has divergent sub-sequences.
I am aware that a divergent sequence can have convergent sub-sequences.
$(-1)^n$ is divergent and has ...
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Given $X = (C[0,1], d_\infty)$ and $f_n \in X$ then for every $K \subseteq X$ compact, $f_n \nsubseteq K$
Given $X = (C[0,1], d_\infty)$ and $f_n \in X$. I want to show that for every compact set $K \subseteq X$, $\{f_n : n \geq 1\} \nsubseteq K$.
Where
$f_n(x) =
\begin{cases}
0, &\quad 0 ...
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2
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Can the monotonicity criterion be dropped from the Abel's Test for Uniform convergence?
(Abel's Test)
Let $\left(f_n\right)$ and $\left(g_n\right)$ be two sequences of real-valued functions on a set $X$. Assume that:
(i) $\sum_n f_n$ is uniformly convergent on $X$.
(ii) There exists $M&...
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1
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62
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I wonder why the author assumed $(f_n)_{n\in\mathbb{N}}$ converges pointwise to a function $f$ on $[a,b]$. ("Calculus I" by Shizuo Miyazima)
I am reading "Calculus I" (in Japanese) by Shizuo Miyazima.
Theorem 6.5
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of continuous functions on $[a,b]$.
Suppose $(f_n)_{n\in\mathbb{N}}$ ...
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0
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104
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Check that $\sum_{n=0}^{\infty}(1-x^2)x^n$ uniformly converges on $[0;1]$
I was asked to check if the series of function $\sum_{n=0}^{\infty}(1-x^2)x^n$ uniformly convergences on $[0;1]$ or not.
This is my first try:
First, I find the partial sum:
$S_n(x)=\sum_{k=0}^{n-1}(1-...
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1
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64
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If $f_n$, f are isometries, then $\underset{n\to\infty}{\lim}\overset{\infty}{\underset{i=0}{\sum}}2^{-i-1}d(f_n(x_i), f(x_i))=0$.
The Problem: Suppose $(X, d)$ is a complete separable metric space, and $\{x_i\}\subseteq X$ is a countable dense subset. Suppose $(f_n)$ is a sequence of isometries of $X$ such that $\underset{n\to\...
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1
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Convergence in sequential Lebesgue spaces.
Consider a strictly increasing sequence $q_0<q_n<q_{n+1}<q<d$ such that $q_n\to q$ as $n\to \infty$ with $d>q$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)...
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1
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62
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check uniformly convergence of $f_n(x)=\begin{cases} nx & \text{ if } 0\leq x \leq \frac{1}{n}\\1& \text{ if } \frac{1}{n}<x\leq 1\\ \end{cases}$
I was asked to check if $f_n(x)=\begin{cases} nx & \text{ if } 0\leq x \leq \frac{1}{n}\\1& \text{ if } \frac{1}{n}<x\leq 1\\ \end{cases}$ uniformly convergence on $\left [ 0;1 \right ]$ by ...
- 121
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112
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Examples of spaces where compact convergence is not equivalent to local uniform convergence.
Let $f_n: X \to Y$ be a sequence of functions where $X$ is a topological space and $Y$ is a metric space. We know that if $X$ is locally compact then local uniform convergence of $f_n$ is equivalent ...
- 473
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1
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81
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Uniform convergence, functions,sequences,subsequences
Let $f_{n}$ be a sequence of continuous functions with $f$ continuous on $\left[0,1\right]$. Prove that if $x_{n}$ is a sequence in $\left[0,1\right]$ such that $\lim_{n\to\infty}x_{n}=x$ for some $x\...
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19
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If a functional sequence converges compactly on a domain, does its subsequence also converge compactly on the same domain?
I have read the proof of Ostrowski's overconvergence theorem (from the Remmert's book 'Classical Topics in Complex Function Theory') and I am confused about one step in the proof.
Let $\sum_{i=0}^{\...
- 1
1
vote
1
answer
43
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A ball in $H^m(\Omega)$ is closed under $L^2$-convergence [closed]
I read it in a paper in which the author aims to claim that if a sequence $\{u_n\}$ is bounded in $H^m(\Omega)$ (here $\Omega$ is a bounded domain), and $u_n$ has a limit $u$ in $L^2$ norm, then $u$ ...
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1
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65
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Lambda function which generates a sequence
Consider this sequence of lambda functions: $\lambda xy. yxy$, $\lambda xyz. zxyz$, $\lambda xyzw. wxyzw$, etc. I would like some lambda function $g$ such that when $g$ is applied to $n$, it returns ...
- 978
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0
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28
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Measurability of a limit "in measure" of a sequence of measurable functions
I am trying to prove the following:
Let $\left(f_n\right)$ be a sequence of measurable functions on a measure space $\left( X, \mathcal{A}, \mu \right)$, and let $f: X \to \mathbb{R}$. Suppose that
$$\...
1
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1
answer
25
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Proving interchange of limiting operations for a sequence of functions dominated by another sequence of functions.
Let $E$ be a borel-measurable set and $f_k, g_k$ two sequences of functions that are defined and measurable on $E, \forall k \in \mathbb{N}$. Also they converge pointwise $\forall x \in E: \lim \...
