Questions tagged [almost-everywhere]

For questions about the concept of "almost everywhere", that is, questions about properties which holds everywhere, except on a set of measure 0. This is involved in probability theory as well as in the case of infinite measure space. Use the appropriate tags to specify the context.

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Is the following characterisation of measurable functions true?

I am self studying measure theory.I measure theory it is often important to check if a function is measurable.If the function is continuous then it is measurable of course.But if the function is not ...
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1 vote
1 answer
52 views

Is almost everywhere equality preserved under integration?

Suppose we have four Lebesgue measurable functions: $$ f_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \qquad f_2: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \\ g_1: \mathbb{R} \...
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  • 301
4 votes
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$L^p_{\text{loc}}$ convergence implies almost everywhere convergence

Suppose $\{f_n\}\subset L^p(B)$ for the unit ball $B\subset\mathbb{R}^n$ converges in $L^p_{\text{loc}}$, i.e. $\int_V\lvert f_n(x)-f(x)\rvert^p dx\rightarrow 0$ for all $V\subset\subset B$. Can we ...
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2 votes
3 answers
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Brownian motion has unbounded variation.

I tried to solve this problem in the following way. Suppose $\{B_t | t\in [0,1]\}$ is our Brownian motion. Define, $$f_n(w) = \sum_{k=1}^{2^n} \bigg|B_{\frac{k}{2^n}}(w)-B_{\frac{k-1}{2^n}}(w)\bigg|.$$...
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1 answer
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Conditional Expectation and Indicator variable

Question Assume that for integrable random variables $X$ and $Y$ we have $E(X|Y)=Y$ and $E(Y|X)=X$. Show that for $x\in \mathbb{R}$ we have $0\leq E[(X-Y)1(Y\leq x< X)]=E[(Y-X)1(X>x,Y>x)]$ $0\...
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1 answer
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a.s. convergence to constant and expectation

Is it true that, if $X_n$ converges a.s. to a constant $c$, then $\displaystyle\lim_{n\to\infty}\mathbb{E}(X_n) = c$? If yes, please, can you prove that? Update: If it is false, is there a condition I ...
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$\frac{1}{N}\sum_{n=1}^{N}X_n Y_n \to pZ \ \ \ \text{a.s.}$

I'm stuck with this exercise... Let $\{X_n\}$, $\{Y_n\}$ r.v. bounded and independent such that $$\frac{1}{N} \sum_{n=1}^{N}X_n \to p \ \ \ \text{a.s.},$$ prove that $$\frac{1}{N}\sum_{n=1}^{N}X_n Y_n ...
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4 votes
1 answer
65 views

How do I show that a continuous function preserves a.s. convergence?

I have the following question: We have $(X_n)_{n},X$ a collection of real valued random variables which are defined in $(\Omega, F, \Bbb{P})$. And $f:(\Bbb{R},B(\Bbb{R}))\rightarrow (\Bbb{R},B(\Bbb{R}...
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2 votes
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For independent Bernouilli RVs, $X_n \to 0$ a.s. iff $\sum \mathbb{P}(X_n=1) < \infty$

I have the following problem. Let $(p_n)$ be a sequence of real numbers in $[0,1]$ such that $p_n\rightarrow 0$. We take $(X_n)$ be a sequence of random variables s.t. $$\Bbb{P}(X_n=0)=1-p_n,~\Bbb{P}(...
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  • 1,284
2 votes
1 answer
93 views

If $f=g$ almost everywhere, $\int f=\int g$

Let $\Omega$ be a measurable set, and let $f: \Omega \to [0, + \infty]$ and $g : \Omega \to [0, +\infty]$ be non-negative measurable functions. Show that if $f(x) = g(x)$ for almost every $x \in \...
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1 vote
1 answer
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Doubts regarding almost uniform convergence

In measure theory I encountered Egorov's theorem which states that if $(X,\mathcal S,\mu)$ is a measure space such that $\mu(X)<\infty$ i.e. $\mu$ is a finite measure.If $(f_n)$ be a sequence of ...
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Application of a result with a stopping time instead of a deterministic time

Setting We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},P)$ with finite time horizon $T$. Assume we are given a result of the following form: Theorem If a ...
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What kind of convergence is understood with almost everywhere convergence?

I am studying Folland's text on measure theory and I am currently on section 2.4. In the beginning of this section he states If $\{f_n\}$ is a sequence of complex-valued functions on a set $X$, the ...
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A topology $\tau$ on $ L([0,1])$ that induces the $almost-everywhere$ convergence on $[0,1]$

Let be $L([0,1])$ the vector space of equivalence classes of measurable function $f:[0,1]\rightarrow \mathbb{R}$ (I don't have more details about the type of equivalence that the exercise is talking ...
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1 vote
1 answer
38 views

convergence almost eveywhere of an uniformly bounded sequence of functions [closed]

Let $(\Omega,\mu)$ be a measure space. Suppose that $(f_n)$ converges almost everywhere to some function $f$ where each $f_n$ belongs to $L^\infty(\Omega)$. Suppose that $\sup_n \|f_n\|_\infty <\...
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3 votes
0 answers
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necessary and sufficient conditions of $\frac{X_n}{a_n}\rightarrow0, \frac{S_n}{a_n}\rightarrow0$ and $\frac{max\{X_1,...,X_n\}}{a_n}\rightarrow0$.

$X_1,...,X_n$ are independent identical random variables of standard normal distribution, $S_n=X_1+...+X_n$, $a_n \uparrow \infty$, try to give the necessary and sufficient conditions of (1)$\frac{...
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Almost sure convergence for absolute value of random variables and inequalities and supremums

I read this article about almost sure inequality and the supremum from Almost sure inequality and the supremum Let $X_n$ be a sequence of random variables, if $\left|X_n \right| \leq Y$ almost surely ...
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1 vote
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Show that the sum of the random variables converges absolutely almost surely for given conditions.

Here is my question and here is my partial solution. I am uncertain if I proved the hint correctly, and I am basically stuck now after proving the hint. I have seen a solution to a similar question ...
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1 answer
41 views

Showing that the absolute value of |X| converges almost surely.

Here is my question and my solutions. I am just curious if this makes sense. If something is wrong, then it will be nice if you can correct it for me or provide another solution. Please and thank you!...
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0 answers
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L2 convergence of weighted iid normal distribution

Let $(X_k)_{k=1}^\infty$ be a sequence of iid random variables distributed $\mathcal{N}(0,1)$. Show that $$S_n=\sum_{k=1}^n \frac{X_k}{2^k}$$ converges in $L^2$ and determine the distribution of the ...
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2 votes
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Having the same law of its Conditional Excpectation implies equality almost surely

Let $X \in L^1(\Omega, F, P)$ r.r.v., $Y \in L^1(\Omega, F, P)$ r.r.v. and $G \subset F$ be a sub-algebra. (r.r.v. Real Random Variable) $X = E[X|G] $ in law $\Rightarrow$ $X = E[X|G]$ a.s. $Y = E[X|...
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1 vote
1 answer
32 views

If the measure of X is infinity and fn converges to 0 a.e. then is it true that fn converges to 0 in measure?

I was thinking of the example in Folland on page 61 i.e. $\mu(\mathbb{R})=\infty$. Let $f_n=n\chi_{[0,1/n]}\rightarrow 0$ a.e. Then $f_n\rightarrow 0$ in measure. My inclination is that this is true ...
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-1 votes
1 answer
57 views

Is it possible that a random variable is greater 0 a.e. and its expectation is still 0? [closed]

I am trying to prove a simple question in probability. It is quite obviously, but somehow I could not prove it. Suppose that a random variable $X > 0$ a.e.. It is possible that ${E}(X)=0$? Thank ...
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0 answers
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Is a function on a product space measurable if it is equal almost surely to a product measurable function?

My question is the following: Consider Borel spaces $(S, \mathcal{S})$ and $(T, \mathcal{T})$, where $\mathcal{S}$ and $\mathcal{T}$ are Borel $\sigma-$algebra of $S$ and $T$ respectively. Consider a ...
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2 votes
0 answers
49 views

Prove that uniform convergence almost everywhere implies convergence almost uniformly?

How can I prove the following result? Uniform convergence almost everywhere implies convergence almost uniformly. I know Egorov's theorem which says pointwise convergence almost everywhere implies ...
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2 votes
1 answer
113 views

almost sure convergence via subsequence arguments

Suppose I have a sequence of random variables $\{X_n\}_{n \in \mathbf{N}}$ such that for every subsequence there exists a further subsequence that converges almost surely to $X$. Can I prove that $X_n ...
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0 answers
17 views

If a function to be maximized is almost everywhere differentiable can we say the derivative=0 condition must hold almost everywhere?

We have a known, increasing function $p:[0,1]\rightarrow[0,1]$. We want to find conditions on functions $R_1(\cdot),R_0(\cdot)$, $R_1(\cdot)$ known to be strictly increasing and $R_0(\cdot)$ known to ...
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  • 1,047
3 votes
0 answers
61 views

Can a property hold almost everywhere and also nowhere?

Let $X$ = {0}. Use the trivial measure on $X$ such that $\mu(A)$ = 0 for all measurable sets in $X$. Let the property $P$ be such that $P$ does not hold at 0. But then there exists a set $N = X$ with $...
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2 votes
1 answer
53 views

$\int_0^1f(t)\phi'(t)dt=-\int_0^1g(t)\phi(t)dt$, for all smooth $\phi\in[0,1]$ implies $f$ is absolutely continuous and $f'=g$ a.e.

I'm trying to solve the following problem. Let $f,g\in L^1[0,1]$ such that for all $\phi\in C^\infty[0,1]$ with $\phi(0)=\phi(1)$, $$\int_0^1f(t)\phi'(t)dt=-\int_0^1g(t)\phi(t)dt.$$ Show that $f$ is ...
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2 votes
1 answer
54 views

Proving $n^{-1/2}\sum_{k=1}^{n}a_{nk}X_k\to0$ a.s

I encountered an exercise while learning the law of large numbers. $\{X_n;n\ge 1\}$ is a sequence of independent and identically distributed random variables.Prove that $EX_k=1,EX_{1}^{2}<\infty \...
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5 votes
1 answer
184 views

Proving $\frac{1}{n} \sum_{k=1}^{n}X_{k}\to 0$ a.s.

$\{X_{n}\}$ is a sequence of independent random variables, $EX_n=0$, and $\sum_{n=1}^{\infty}n^{-(r+1)}E(|X_n|^{2r})<\infty$. Proving $\frac{1}{n} \sum_{k=1}^{n}X_{k}\to 0$ a.s. and $r>1$ I ...
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2 votes
1 answer
85 views

Showing that a martingale $Y_k$ does not converge almost surely.

Let $X_i$ be iid with $$\mathbb{P}(X_i=1)= \mathbb{P}(X_i= -1) = \frac{1}{2i}, \mathbb{P}(X_i=0)=1-\frac{1}{i},$$ where $i=1,2,...$ And define $Y_1=X_1$ and for $k\geq2$ $$Y_k= \begin{cases} X_k, \...
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4 votes
1 answer
57 views

Almost sure convergence of rescaled nondecreasing sequences of random variables

Let us consider a sequence $(S_n)_n$ of $L^2$ random variables. Assume: $S_n \le S_{n+1}$ almost surely $S_n \to_{n \to \infty} +\infty$ almost surely $\frac{S_n}{\mathbb{E}[S_n]} \to_{n \to \infty} ...
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0 answers
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Showing different definitions of almost sure convergence are equivalent.

There are a couple different equivalent definitions of almost sure (a.s.) convergence: $\forall \varepsilon>0\quad P(\liminf_{ n\uparrow \infty}\{\|X_n-X\|\leq\varepsilon\})=1$ $\mathbb{P}(\{\...
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  • 7,897
0 votes
1 answer
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Proof verification of functions being the same almost everywhere

Let ($X, \mathscr{F}, \mu)$ be a measure space and, $T:X \to X$ $\mathscr{F}/\mathscr{F}$ measurable map such that: $\mu(C)=\mu(T^{-1}(C))$ for all $C\in \mathscr{F}$, and let $u\in\mathscr{L}^1(\mu)....
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0 votes
1 answer
52 views

L2 convergence for a simple function approximation

Consider the problem on the picture. I am struggling with part (b) of the excercise. I have managed to show that we have $L^1$ convergence, but I am unable to show $L^2$ convergence. Does anyone have ...
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0 votes
2 answers
46 views

$\sqrt{n} \mathbb{1}_{[0,\frac{1}{n}]}$ converges simply almost everywhere to $0$. [duplicate]

I’m currently reading a book about Fourier series and I stumbled about a statement that won’t leave me alone (It’s not in relation with a Fourier series though). The statement is as follows: $\sqrt{n} ...
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0 votes
0 answers
81 views

Fundamental theorem of calculus for monotone function

Assume $F:(0,1] \to \mathbb{R}$ is given by $F(x) = \int_0^xf(u) du$ where $f:(0,1) \to \mathbb{R}$ is a nonincreasing (not necessarily continuous) function (note that F is real-valued so it is ...
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  • 908
0 votes
0 answers
13 views

Does almost sure convergence of function imply almost sure convergence of random variable

Let $g_n(x)$ be a sequence of random functions. Suppose that $g_n(x) \to g(x)$ almost surely for all $x$ and $g(x)$ is a fixed function. Is it true that for any random variable $X$, we also have $g_n(...
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0 votes
1 answer
184 views

Borel-Cantelli-Lemma and almost sure convergence

Suppose it holds: $P(|S_n|> \varepsilon )\leq \frac{1}{2^n \varepsilon^2} $, where $S_n $ is sequence of random variables. Furthermore it holds: $\sum_{n \geq 1} P[|S_n|> \varepsilon ] <\...
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  • 57
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0 answers
19 views

Definition of the space $\mathcal{R}_2([\mathbb{R},\mathbb{C}])$

I'm studying the Fourier series on Zorich II book. In this context he introduces the vector space $\mathcal{R}_2([\mathbb{R},\mathbb{C}])$ of function that are "locally square-integrable, as ...
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1 vote
1 answer
82 views

A generalization of the law of large numbers.

Let $X\subset \mathbb{R}^m$ be compact. Let $X_1,\dots,X_n \sim_{iid}\mathcal{U}(X)$. By the law of large numbers, we have $P(\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n(f(X_i)-E[f(X_i)])=0)=1$ for all $...
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  • 732
2 votes
1 answer
51 views

$n^{\alpha}U_{(1)} \to 0$ almost surely for any $\alpha < 0.5$

Let $U_i$ be a sequence of identically distributed independent random variables with following density function $f(x) = 2x \ \ x \in[0,1]$ and $0$ everywhere else. I need to show $n^{\alpha}U_{(1)} \...
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0 votes
0 answers
41 views

Can we prove almost everywhere absolute continuity?

This problem arises in my research in the field of information theory, any reference to some similar work, any comment about style or imprecisions are most welcome. Let $\mathcal X$ be some space and $...
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  • 4,135
5 votes
1 answer
107 views

Does squeeze theorem apply for almost sure convergence

Suppose we are given that $X'_n \leq Z_n \leq X_n$ for random variables $X'_n,Z_n, X_n$. If we are told that $X_n,X'_n$ converges almost surely to some random variable $Y$, can we conclude that $Z_n \...
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  • 109
0 votes
1 answer
37 views

a.s. convergence of two sequences

Assume we have two sequences of random elements $X_{n}$ and $Y_{n}$, taking values from some Hilbert space $S$ and defined on the same probability space. Assume that $$ X_{n}\overset{a.s.}{\to} a $$ ...
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  • 129
0 votes
0 answers
26 views

Difference between equality a.s. and equality in distribution: dice example.

Suppose we model two dice by the random variables $X_1$, $X_2$. Then we have $X_1\stackrel{d}{=}X_2$. Which means that they have the same probability distribution but for each realization they could ...
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3 votes
1 answer
162 views

Convergence of Stationary random variables

We have a stationary sequence of random variables $X_{j}:j\geq 0$ and let $D$ be a Borel subset of $\mathbb{R}^{d}$. For each n, let $Y_{n}$ be the number of indices $i \in \{0,1, \ldots, n-d\}$ such ...
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  • 204
0 votes
1 answer
50 views

Can we construct a counterexample that satifies the conditions?

Let $(X,\mathscr{A},\mu)$ be a measure space. The Riesz Theorem in real analysis shows that, for $A\in\mathscr{A}$, functions $f,f_{n}:A\to\mathbb{R}$ are $\mu$-measurable, if sequence $\{f_n\}$ is $\...
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  • 3
3 votes
0 answers
121 views

A.e. $L^p$ convergence implies a.e. convergence along sub-sequence

The book on Bochner spaces that I am currently looking at contains the following theorem: Let $\tilde{u}:[0,T] \to L^p(a,b)$ be Bochner measurable for some $1\leq p< \infty$. Define $$u: [0,T] \...
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