# 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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### 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. ...
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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 ...
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### 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. ...
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### 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 $f$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $([0,1], \... • 1,381 1 vote 2 answers 92 views ### Measurability and almost sure convergence, going from epsilon to 1/j I am trying to understand the definition of almost sure convergence. For this one needs to check if there are no measurability issues. By transforming the definition of convergence to set theory ... 0 votes 1 answer 42 views ### Does additivity of probability imply almost-sure pairwise disjointedness? Suppose we have a probability measure$P$and a countable collection of sets$\{ E_k\}_{k=1}^{\infty}$from the corresponding$\sigma$-algebra. If $$P\left( \bigcup_{k=1}^{\infty} E_k \right) = \sum_{... • 1,876 2 votes 1 answer 47 views ### measure theory: g continuous a.e., f_n \to f a.e., does g \circ f_n \to g \circ f a.e.? Question: Let (f_n)_{n \geq 1}, f, and g be Borel measurable functions from \mathbb{R} \to \mathbb{R}. Suppose g is continuous almost everywhere and f_n \to f almost everywhere, does g \... • 451 2 votes 0 answers 32 views ### For a Wiener process, when can one exchange “for all t” and “almost surely”? Certain local properties of the Wiener process W_t are quick to prove at t = 0, for instance: almost surely W_t is monotonous on no interval beginning at t = 0; almost surely W_t is not ... • 514 0 votes 0 answers 56 views ### Almost everywhere differentiable functions: what can we do? Integration by parts? Let A be an open measurable set and A_0 is a subset of A which has measure zero. Let g and f be almost everywhere differentiable function defined on A. What is the impact of "almost ... • 203 1 vote 0 answers 38 views ### Does separate almost everywhere continuity imply joint almost everywhere continuity? My question is almost the same as this question. Let f(\cdot, \cdot): R^2 \rightarrow R. Suppose that: (a) Fix any y \in R, f(\cdot, y) is continuous almost everywhere. (b) Fix any x \in R, f(x, \... • 1,473 7 votes 1 answer 341 views ### If random variables X and Y are equal in distribution, then there exists a measurable function f such that X(\omega)=Y(f(\omega)) a.s. Let X and Y be two identically distributed random vectors in \mathbb{R}^{d} defined on the same underlying probability space (\Omega,\mathcal{A},P). Suppose that P is non-atomic. Does there ... • 2,473 1 vote 0 answers 82 views ### Showing that a function is nonnegative almost everywhere on (0,1) Let f be an integrable function in (0,1). Suppose that$$\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 (... 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}... • 329 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 ... 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 ... • 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}}^... 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 ... 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}. ... • 430 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, $$where ... • 1,484 0 votes 0 answers 38 views ### Almost sure bound for maximal of Gaussian R.V. The following two questions come from some steps of my thought about solving an exercise. Thanks for help. Let X_1, X_2, ..., X_n, ... be i.i.d N(0,1) random variables. Define M_n:=\max_{1\leq i ... • 2,307 0 votes 0 answers 54 views ### Equality of two sets in measure theory, with finite function for almost everywhere I'm currently solving exercise 10 of chapter 2 of Stein's Real Analysis. The problem is given as follows. What I'm struggling to is the first part :$$\bigcup_{k = -\infty}^{\infty}F_k = \{x : f(x) &... • 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 $on ... • 1,484 1 vote 0 answers 34 views ### Defining convergence of a sequence of random variables. I am trying to understand ideas like Almost sure convergence and convergence of sequences of random variables. I am trying to prove: Given an infinite sequence of random variables$X_i,i\in\mathbb{N}$... • 151 4 votes 1 answer 67 views ### Waiting times between record observations Let$\{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 \...
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 ...