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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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We get $L^2$ convergence, but do we get a.s. convergence?

Assume you have a sequence of independent Bernoulli random variables $X_i$ each with probability $p_i$. Let $c_i$ be a sequence of real numbers and $ m,M$ be a real numbers such that $0 < m <c_i&...
user394334's user avatar
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If $f_n \rightarrow f$ almost everywhere and all $f_n$ are measurable, how can I redefine $f$ on a null set to make it measurable?

I am self-studying real analysis out of Folland, and I am confused at the beginning of his proof of the Dominated Convergence Theorem. It reads as follows, where all $f_n$ map from measure space $(X,M,...
Lightbulb's user avatar
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1 answer
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Alomost sure convergence

I was studying the various types of convergence and I met a question for almost sure convergence. Consider random varibales $X_n,X:\Omega\to\mathbb{R}$, $\Omega$ is a probability ($\mathbb{P}$) space. ...
R-CH2OH's user avatar
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Definition of integral of an almost everywhere non-negative, measurable function

Let $(X, \mathcal{M}, \mu)$ be a measurable space and $(X, \overline{\mathcal{M}}, \overline{\mu})$ be its completion. Let $f \colon X \longrightarrow \overline{\mathbb{R}}$ be an almost everywhere ...
user1063822's user avatar
1 vote
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Lebesgue measure simple property. Seems intuitevely true but can't formalise a proof.

Consider the usual Lebesgue measure in $\mathbb R^n$ and let $f,g \colon \mathbb R^n \to \mathbb R$ be two arbitrary functions that satisfy $$ | \{ x \in \mathbb R^n : g(x) \neq h(x) \} | > 0 \quad ...
Temirbek Alikhadzhiyev's user avatar
5 votes
1 answer
92 views

Convergent sequences under different probability measures

What is the „intuitive” reason behind the following statements... Let $\left(X_{n}\right)_{n}$ be a sequence of random variables. Let us assume $\mathbf{Q}\ll\mathbf{P}$, i.e. the $\mathbf{Q}$ ...
Kapes Mate's user avatar
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3 votes
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Understanding Proof of Proposition 2.2.5 from Measure Theory by Donald Cohn

Overview I am self-studying Donald Cohn's Measury Theory. I have some questions about his proof of Proposition 2.2.5, and I would like to confirm if my attempt to understand his proof is correct. Here ...
Beerus's user avatar
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0 votes
1 answer
34 views

Convergence in measure and almost uniform Cauchy convergence almost everywhere

I,m trying to prove the following: Let $(X, \Omega, \mu)$ be a measure space and $f,f_n: \Omega \to \mathbb{K}$, $n \in \mathbb{N}$ be measurable functions where $\mathbb{K} = \mathbb{R}$ or $\mathbb{...
Kham Bodrogi's user avatar
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1 answer
30 views

Convergence in $L^\infty$($\Omega$) and almost everywhere

I have a question about the difference between the convergence in $L^\infty$ and convergence almost everywhere. Precisly, let $\mu(\Omega) < \infty$, $f_n \rightarrow f$ in $L^1(\Omega)$, then ...
Annabelle's user avatar
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Is my construction almost-surely a conditional Poisson random variable?

Suppose a continuous random variable $$L \sim F_{L}$$ where $$Pr[L = 0] = 0$$ even though $$L = 0$$ exists for some countable subset of the outcome space. I am considering the existence of a variable $...
Galen's user avatar
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1 answer
69 views

Understanding the term almost everywhere in measure theory

I am trying to understand the term "almost everywhere" from measure theory correctly. So given two extended real-valued integrable functions $f, g: X \rightarrow \bar{\mathbb{R}}$ with $$\...
guest1's user avatar
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1 answer
31 views

Existence of a limit from limsup and liminf

Consider a sequence of random variables $(p_n)$ such that: (1) $\Pr\bigg(\limsup_{n \to \infty} p_n\leq U\bigg)=1$ (2) $\Pr\bigg(\liminf_{n \to \infty} p_n\geq L\bigg)=1$ where $L,U$ are real numbers. ...
Star's user avatar
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1 answer
184 views

Confusion on "almost everywhere defined" function in $L^1$ space.

The following are excerpt from folland This proposition shows that for the purposes of integration it makes no difference if we alter functions on null sets. Indeed, one can integrate functions $f$ ...
juekai's user avatar
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2 votes
1 answer
105 views

Is there a better or more accepted way to say ``for almost any $t$"

Is "almost any $t$" valid mathematical terminology? I am familiar with almost surely, almost everywhere, etc. Here is the situation in which I am interested. Suppose we have two (discrete ...
Jack's user avatar
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1 answer
209 views

The quantifiers in the definition of almost sure convergence

Let $X_1,X_2,X_3,\dots$ be a sequence of random variables, defined on the probability space $(\Omega, \mathcal{F} , \mathbb{P})$. We say that this sequence converges almost surely to random variable $...
S.H.W's user avatar
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2 votes
1 answer
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Last hitting time of $0$ of a Brownian motion in $[0,1]$ is a.s. less than one?

Let $L=\sup\{t\in[0,1]:B_t=0\}$. How does one show that $L<1$ a.s.? I know that that $B_t$ has infinitely many zeros in [0,1], (in fact, in $(0,\epsilon)$). I also know that the probability of $B_1=...
J.R.'s user avatar
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1 vote
1 answer
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Convergence of sum of sine of Gaussian

Suppose $\alpha>0$, $\xi_n$ are independent standard gaussians, $S_n = \sum_{k=1}^{n} \sin(\frac{\xi_k}{k^\alpha})$, (a) Find all $\alpha$ such that $S_n$ converges with probability 1. (b) If $\...
喵喵露's user avatar
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1 vote
1 answer
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Pointwise convergence up to a subsequence and Sobolev spaces

Let $u_m\to u$ in $W^{k,p}(\Omega)$ for some domain $\Omega\subset\mathbb{R}^n$. I feel like, up to a subsequence, i can say that $D^\alpha u_m\to D^\alpha u$ a.e. for every $|\alpha|\leq k$. The ...
DagunDagun's user avatar
4 votes
2 answers
97 views

Strong law of large numbers for $\mathrm{Bin}(n, p_n)$ variables

Massive edit to simplify the central question Suppose $X_n\sim \mathrm{Bin}(n, p_n)$ be a collection of independent random variables such that $np_n\to \infty$. Can we say that $Y_n:=X_n/np_n\to 1$ ...
Landon Carter's user avatar
1 vote
1 answer
120 views

Does the sequence converge on $\mathbb{R}$?

Does the sequence $f_n(x) = \sin^n(x) + \cos^n(x)$ converge on $\mathbb{R}$? a) almost everywhere b) according to Lebesgue measure My solution: a) $\lim\limits_{n \rightarrow \infty} (\sin^n(x) + \cos^...
rebrom1's user avatar
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2 votes
1 answer
81 views

If $X$ and $Y$ have same distributions and $X \leq Y$ almost surely, does $X=Y$ almost surely?

If $X$ and $Y$ have same distributions and $X \leq Y$ almost surely, does $X=Y$ almost surely? Here is a specific problem I met. Assume $f(\omega)$ be a real-valued random variable which is defined on ...
R-CH2OH's user avatar
  • 351
5 votes
1 answer
96 views

integral evaluating to a quantity 'almost everywhere' in $\mathbb{R}^2$

Let $g:[0,1]^2 \to [0,1]$ be a measurable function. Suppose $\int\limits_0^1 g(x,t)g(y,t)dt = A$ holds for almost every $(x,y)\in [0,1]^2$ then prove that $\int\limits_0^1 g^2(x,t)dt = A$ for almost ...
Snowflake's user avatar
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1 vote
1 answer
72 views

Proof of a pointwise convergence of characteristic functions

Let $(X, \mathcal{F},p)$ be a probability space and, for a fixed $A \in \mathcal{F}$, let $\chi_{A}(x)$ denote the characteristic function of the set $A$, that is, $\chi_{A}(x) = 1$ if $x \in A$ and ...
InMathweTrust's user avatar
1 vote
1 answer
68 views

Finding a constant for the mean value theorem

If $ g(x) \geq 0 $ and, I want to calculate $ \int f(x)g(x) dx $ for some $ f(x)$. Can I find a constant $c$ satisfying $$ \int f(x)g(x) dx = \int cf(x) dx \text{ ? }$$ or even can I claim that $ c= ...
Lee's user avatar
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1 vote
1 answer
94 views

Combining Convergence in Distribution and Almost Surely

Suppose I know that $\frac{1}{\sqrt{m}}X(mt)\xrightarrow[m\to\infty]{d} B(t)$ where $B$ is Brownian motion, and that $\alpha(mt)/m \xrightarrow{a.s.} 0$. Then I am trying to prove the following ...
user1598's user avatar
  • 395
0 votes
1 answer
26 views

A.S. convergence of i.i.d random variables [closed]

Let $ (X)_{n\in\mathbb{N}}$ be i.i.d random variables with $P(X_j=1/2)=P(X_j=3/2)=1/2$. Also $Z_n:= \prod_{i=1}^n X_i$. Determine $\lim \limits_{n \to \infty}Z_n $ a.s. I'm relatively new to this so I'...
Ice's user avatar
  • 13
0 votes
0 answers
29 views

Use Markov inequality to prove that a positive random variable is null a.s

I would like to show that if the expectation of a positive random variable $X$ (integrable) is $0$ then $X$ is null almost surely. I thought on a proof and would like to know if it is correct, it uses ...
G2MWF's user avatar
  • 1,381
0 votes
1 answer
86 views

A problem about Lebesgue integral convergence

Suppose that $\{E_n\}_n$ is a sequence of Lebesgue measurable subset in $\mathbb R$ satisfies $\displaystyle\lim_{n\to \infty}\mu(E_n)=0$. $\{f_n\}_n$ is a Lebesgue measurable function sequence on $\...
QIRUN CONG's user avatar
0 votes
1 answer
108 views

An exemple that convergence in probability does not imply convergence almost surely

In my lecture notes, there is the following example to study in order to show that convergence in probability does not imply convergence almost surely. We consider the measurable space $([0,1], \...
G2MWF's user avatar
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1 vote
2 answers
92 views

Measurability and almost sure convergence, going from epsilon to 1/j

I am trying to understand the definition of almost sure convergence. For this one needs to check if there are no measurability issues. By transforming the definition of convergence to set theory ...
Fledermaus's user avatar
0 votes
1 answer
42 views

Does additivity of probability imply almost-sure pairwise disjointedness?

Suppose we have a probability measure $P$ and a countable collection of sets $\{ E_k\}_{k=1}^{\infty}$ from the corresponding $\sigma$-algebra. If $$P\left( \bigcup_{k=1}^{\infty} E_k \right) = \sum_{...
Galen's user avatar
  • 1,876
2 votes
1 answer
47 views

measure theory: $g$ continuous a.e., $f_n \to f$ a.e., does $g \circ f_n \to g \circ f$ a.e.?

Question: Let $(f_n)_{n \geq 1}$, $f$, and $g$ be Borel measurable functions from $\mathbb{R} \to \mathbb{R}$. Suppose $g$ is continuous almost everywhere and $f_n \to f$ almost everywhere, does $g \...
HIH's user avatar
  • 451
2 votes
0 answers
32 views

For a Wiener process, when can one exchange “for all $t$” and “almost surely”?

Certain local properties of the Wiener process $W_t$ are quick to prove at $t = 0$, for instance: almost surely $W_t$ is monotonous on no interval beginning at $t = 0$; almost surely $W_t$ is not ...
Olius's user avatar
  • 514
0 votes
0 answers
56 views

Almost everywhere differentiable functions: what can we do? Integration by parts?

Let $A$ be an open measurable set and $A_0$ is a subset of $A$ which has measure zero. Let $g$ and $f$ be almost everywhere differentiable function defined on $A$. What is the impact of "almost ...
Eryna's user avatar
  • 203
1 vote
0 answers
38 views

Does separate almost everywhere continuity imply joint almost everywhere continuity?

My question is almost the same as this question. Let $f(\cdot, \cdot): R^2 \rightarrow R$. Suppose that: (a) Fix any $y \in R, f(\cdot, y)$ is continuous almost everywhere. (b) Fix any $x \in R, f(x, \...
Canine360's user avatar
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7 votes
1 answer
341 views

If random variables $X$ and $Y$ are equal in distribution, then there exists a measurable function $f$ such that $X(\omega)=Y(f(\omega))$ a.s.

Let $X$ and $Y$ be two identically distributed random vectors in $\mathbb{R}^{d}$ defined on the same underlying probability space $(\Omega,\mathcal{A},P)$. Suppose that $P$ is non-atomic. Does there ...
möbius's user avatar
  • 2,473
1 vote
0 answers
82 views

Showing that a function is nonnegative almost everywhere on (0,1)

Let $f$ be an integrable function in $(0,1)$. Suppose that $$\int_{0}^{1} fg dx \geq 0$$ for any nonnegative continuous function $g:(0,1) \to \mathbb{R}$. Prove that $f \geq 0$ almost everywhere in $(...
EricBurkholder's user avatar
3 votes
1 answer
100 views

Is convergent sequence in $L^2$ almost surely bounded?

Let $(f_n)_{n\ge 1}$ be a sequence of function that converges to some $f\in L^2(\mathbb{R})$, i.e., $||f_n-f||_{L^2(\mathbb{R})}=0$. Clearly the sequence of norms $(||f_n||_{L^2(\mathbb{R})})_{n\ge 1}$...
user107678's user avatar
1 vote
1 answer
47 views

Conditional probability when the given outcome has probability $0$

Consider two different random variables on $\{0,1\}^{\mathbb N}$, i.e. the set of binary sequences. The first random variable $X_1$ has a distribution defined by letting each of its digits be chosen ...
Franklin Pezzuti Dyer's user avatar
3 votes
2 answers
143 views

Show $\int _{0} ^{\infty} \frac{|f(x)|}{|x-a|^b} dx <\infty$ for a.e. $a\in \mathbb{R}$ if $f$ Lebesgue integrable on $\mathbb{R}$, $b\in (0,1)$

The question: Let $f$ be a Lebesgue integrable function on $\mathbb{R}$ and let $b\in (0,1)$. Prove that $\int _{0} ^{\infty} \frac{|f(x)|}{|x-a|^b} dx <\infty$ for a.e. $a\in \mathbb{R}$. My ...
Thmyem's user avatar
  • 129
1 vote
1 answer
202 views

Weak convergence and almost everywhere convergence on $H_0^1({\mathbb{R}}^N)$

I am seeking a clarification to this question on almost convergence in Sobolev spaces. The essence of the answer there was to prove that if a sequence $(u_n) \rightharpoonup u$ in $H_0^1({\mathbb{R}}^...
IsomorphicBunny's user avatar
1 vote
0 answers
130 views

Separable and diffuse probability space is standard

Let $(X,\mathcal{N},\mu)$ be a complete probability space, and assume that the $\sigma$-algebra $\mathcal{N}$ has no atoms, that is, for every set $A \in \mathcal{N}$ with $\mu(A) > 0$ there is a ...
Henrique Augusto Souza's user avatar
0 votes
1 answer
112 views

Show that $\forall\varepsilon>0$, there exists a bounded continuous function $g(x)$ on $E$, such that $m(\{x\in E\mid f(x)\neq g(x)\})<\varepsilon$.

Let $E$ be a measurable subset of $\mathbb{R}^n$, and $mE<\infty$. $f(x)$ is a bounded measurable function almost everywhere. I want to find such a function $g$. For $n=1$, i.e., $E\in\mathbb{R}$. $...
一団和気's user avatar
1 vote
2 answers
672 views

Are $L^p$ functions continuous almost everywhere?

Let $1 \le p < +\infty$ and define the Lebesgue space as the quotient space $$ L^p(X):= \left\{f\text{: }X\rightarrow \mathbb{R} \mid \int_X \left|f\right|^p d\mu <+\infty\right\}/\sim, $$ where ...
mathslover's user avatar
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0 votes
0 answers
38 views

Almost sure bound for maximal of Gaussian R.V.

The following two questions come from some steps of my thought about solving an exercise. Thanks for help. Let $X_1, X_2, ..., X_n, ...$ be i.i.d $N(0,1)$ random variables. Define $M_n:=\max_{1\leq i ...
Sam Wong's user avatar
  • 2,307
0 votes
0 answers
54 views

Equality of two sets in measure theory, with finite function for almost everywhere

I'm currently solving exercise 10 of chapter 2 of Stein's Real Analysis. The problem is given as follows. What I'm struggling to is the first part : $$\bigcup_{k = -\infty}^{\infty}F_k = \{x : f(x) &...
jason 1's user avatar
  • 769
0 votes
2 answers
44 views

Condition on Sobolev function on the boundary of a domain

I encountered the following kind of a condition when studying boundary value problems: Let $\Omega \subset \mathbb{R}^2$ be a domain. Let $\varphi \in H^1 (\Omega)$ be such that $\varphi \ge 0 $ on ...
mathslover's user avatar
  • 1,484
1 vote
0 answers
34 views

Defining convergence of a sequence of random variables.

I am trying to understand ideas like Almost sure convergence and convergence of sequences of random variables. I am trying to prove: Given an infinite sequence of random variables $X_i,i\in\mathbb{N}$ ...
zak zaki's user avatar
  • 151
4 votes
1 answer
67 views

Waiting times between record observations

Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. random variables with a continuous CDF. Let $V_1:=\min \{n\in\mathbb N\, \vert \,X_n>X_1\}$. Let $V_{r+1}:=\min \{ n\in \mathbb N \,|\, n>V_r \...
Sam Wong's user avatar
  • 2,307
0 votes
1 answer
87 views

Is this property of differentiable almost everywhere functions true?

Consider a function $g:\mathcal{I}\to\mathbb{R}$ defined in an open interval $\mathcal{I}$ of the real line and therein differentiable almost everywhere. Is it true that all points in which $g$ is ...
Mathgineer94's user avatar

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