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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.

This is a concept in probability and measure theory. Something is said to hold almost everywhere if the set on which it does not hold has 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.

515 questions
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Measurability of an a.e. pointwise limit of measurable functions.

Suppose that $(f_n)_n$ is a sequence of measurable functions on a set $E$ and that $f_n \to f$ a.e.on $E$. Does this imply that $f$ is measurable? I know that pointwise limit of measurable function is measurable. But here we only have convergence…
user145993
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20
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Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ converges a.e. to the function $f$ if $f_{n} \to…
möbius
  • 2,573
13
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1 answer

Why do we need to define Lebesgue spaces using equivalence classes?

When we define an $L^{p}$ space for $1\leq p \leq \infty$, we say elements of this space are equivalence classes of functions which are equal almost everywhere and $$ \int|f|^{p} dx < \infty $$ Why can we not say elements are functions which…
Monty
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If $ \int fg = 0 $ for all compactly supported continuous g, then f = 0 a.e.?

I was wondering whether the following statement is true and, if so, how it can be shown: If $ f \in L^{1}_{Loc}(\mathbb{R}^n) $ and if for all compactly supported continuous functions $ g: \mathbb{R}^n \to \mathbb{C} $ we have that the Lebesgue…
Apollo13
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12
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1 answer

Lipschitz continuity implies differentiability almost everywhere.

I am running into some troubles with Lipschitz continuous functions. Suppose I have some one-dimensional Lipschitz continuous function $f : \mathbb{R} \to \mathbb{R}$. How do I prove that its derivative exists almost everywhere, with respect to the…
arriopolis
  • 668
10
votes
1 answer

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let $\delta_0$ be the Dirac law with mass at $0$.…
PAM
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3 answers

If X and Y are equal almost surely, then they have the same distribution, but the reverse direction is not correct

Show that if two random variables X and Y are equal almost surely, then they have the same distribution. Show that the reverse direction is not correct. If $2$ r.v are equal a.s. can we write $\mathbb P((X\in B)\triangle (Y\in B))=0$ (How to…
derivative
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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 exist a measurable function $f:\Omega\to \Omega$…
möbius
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7
votes
1 answer

If $f$ takes every value at most $k$ times, then f is differentiable almost everywhere.

I am stuck at the following problem, I got in an old question paper (real analysis). Let $k>0$ be a natural number and Let $f$ be a continuous function on real line such that $f$ takes any value at most $k$ times. Show that $f$ is differentiable…
Raghav
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7
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1 answer

Dominated a.e. convergence implies almost uniform convergence

Let $(f_n)$ be a sequence of measurable functions that converges almost everywhere to a measurable function $f$. Assume that there is an integrable function $g$ such that $|f_n|\leq g$ for all $n$ almost everywhere. Show that $(f_n)$ converges…
user313212
  • 2,306
7
votes
3 answers

almost everywhere Vs. almost sure

I'm reading a book about measure theory and probability (first chapter of Durret's Probability book), and it's starting to switch between the terms "a.e." and "a.s." in different contexts. I'm becoming confused about their meanings. What's the…
user3585775
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2 answers

Under what condition can converge in $L^1$ imply converge a.e.?

Let $f_n$ be a sequence of Lebesgue measurable functions on $R^d$. Suppose you have an estimate of the form $\int_{R^d}\left|f_n\right|\le c_n$ where $c_n \downarrow 0$. Can you conclude that $f_n\to 0$ a.e.? If not, what additional conditions on…
Sherry
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Boundedness in $L^2$ implies almost everywhere convergence?

Let $\Omega\subset \mathbb{R}^N$ be a bounded domain and $\{u_n\}\subset L^2(\Omega)$ a non-negative sequence. Suppose that $$ \int_\Omega u_n^2dx=\int_\Omega u_ndx, \quad \quad \forall n=1,2,3,\cdots. \tag{1} $$ By (1) and Holder inequality, one…
teh
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2 answers

On the definition on almost sure convergence

I am a bit confused about the definition of almost sure convergence. Of course, we say that $X_n \underset{n \to \infty}{\longrightarrow} X$ almost surely if $$ \mathbb{P} \big( \{ \lim_{n \to \infty} X_n = X \} \big) = 1 $$ The almost sure event of…
Fran712
  • 467
6
votes
2 answers

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 a metric dynamical system…
R-CH2OH
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