Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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1answer
38 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
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0answers
53 views

Why $\int f_0 \mbox{d}(F_0+F_1)=\int f_0 \mbox{d}F_0+\int f_0 \mbox{d} F_1=1$ should be true?

A measure $\mu$ dominates another measure $\nu$ whenever $\mu=0$ implies $\nu=0$. If I would like to take the integral of a measurable function $f_0$, say the density function of the probability ...
0
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1answer
34 views

Converge in measure implies converge a.e if $f_n$ are monotone

Let $\{f_n\}$ be monotone sequence of functions such that $f_n$ converges in measure to $f$. Is it true that $f_n$ converges to $f$ a.e$?$ I am sure it has a sub-sequence that converges to $f$ a.e. ...
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0answers
49 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
7
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1answer
243 views

Notation in Terry Tao's exposition on the PNT

The exposition I'm talking about can be found here (page 6): http://www.math.ucla.edu/~tao/preprints/Expository/prime.dvi Essentialy, Tao proves the prime number theorem in the elementary way, ...
2
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2answers
92 views

Do we need the $f,g \geq 0$ condition for $\int f \ d\mu = \int g \ d\mu$?

My lecture notes state the following corollary: Let $f,g \in \mathcal M_\bar{{\mathbb R}}$ (that is, numerical measurable functions), $f=g$ $\mu$-almost everywhere and $f,g \geq 0$. Then $\int f \ ...
12
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2answers
177 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
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1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
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0answers
52 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
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2answers
28 views

Prove that for $μ$-almost every $x ∈ X$ $−1 ≤ \liminf f_n(x) ≤ \limsup f_n(x) ≤ 1$.

Let ${f_n}$ be a sequence of measurable functions on a measure space $(X, M, μ),$ and suppose that $\sum_{n = 1}^{\infty}μ\{x∈X :|f_n(x)|>1\}<∞.$ Prove that for $μ$-almost every $x ∈ X$ $−1 ≤ ...
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2answers
27 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
3
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0answers
26 views

Transition kernel that is not Markov

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be two measurable space. A transition kernel $K$ is a function $K : X \times \mathcal{G} \to \overline{\mathbb{R}}_+$ suche that $K(\cdot,B)$ is measurable ...
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0answers
33 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
0
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1answer
24 views

product measure of weak (-*-) converging signed measure is weak-(*)-converging?

Assume that we have a sequence of signed measures $\mu_n$ on $[0,1]$ that converge weak(-*) to $\mu$, that means: For all continuous and bounded functions $f:[0,1] \rightarrow R$ we have $\int_0^1 ...
4
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1answer
34 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
2
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1answer
35 views

Clarification about a basic proposition about measurable functions

I am making my way through "Linear Functional Analysis" by Bryan P.Rynne and Martin A.Youngson (second edition). Given a measure space $(X,\Sigma ,\mu )$ we define a function $f$ to be measurable if ...
0
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1answer
55 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
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1answer
40 views

Is the diagonal measurable

Suppose we have $X=Y=[0,1]$, $\lambda$ the Lebesgue measure on $[0,1]$, and $\nu$ the counting measure on $[0,1]$. Show that the diagonal $\Delta=\{(x,x):x\in X\}$ is $\lambda\times\nu$-measurable. I ...
2
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1answer
29 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
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0answers
41 views

von Neumann Algebras and measures

I read that any abelian von Neumann algebra is isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-commutative measurable ...
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0answers
30 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: Let $E$ be a measurable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. Suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
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0answers
25 views

Measure in dual group - Kirillov theory

Let $G$ be a nilpotent connected, simply connected lie group. With the orbit method Kirillov describes the classes of equivalence of all irreducible unitary representations. Hence one identifies the ...
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0answers
82 views

Exercise on Radon measures, constructing a convergent sequence

Let $\mu$ be a Radon measure on $\mathbb{R}^n$ such that $\mu(B(0, s)) > 0$ for all $s > 0$ and suppose that $$C = \limsup_{s\ \downarrow\ 0}\frac{\mu(B(0, 2s))}{\mu(B(0, s))} < ...
0
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1answer
38 views

When does the convergence of the regularization of a function is decreasing?

Hi everyone: Let $\theta(x)$ equal $k\exp\left(-\frac{1}{1-\|x\|^2} \right)$ if $\|x\|<1$, and equal $0$ if $\|x\|\geq1.$ Here $\|\cdot\|$ designates the Euclidean norm in $\mathbb{R}^n$, and the ...
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0answers
57 views

Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
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0answers
53 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
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1answer
45 views

Union of Increasing Sequences of Monotone Classes is not a Monotone Class.

In my text, we define a Monotone Class $\mathcal{M}$ of a non-empty set $X$ to be a collection of subsets of $X$ that is closed under monotone limits: that is, $(1)$ if $A_{i} \uparrow A$ with $A_{i} ...
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2answers
49 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
1
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2answers
30 views

Approximation in non-compact interval

Suppose that $f$ is a continuous function defined on a interval $I\subseteq \Bbb R$. (a) If $I=[0,1]$ and $\epsilon \gt0$ is given show that there are finitely many constants $a_k$, $1\le k \le n$, ...
0
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0answers
20 views

Non-atomic measure has a dense range

Suppose $(X,\mathcal{M},\mu)$ is a non-atomic measure space--i.e., it has no atoms [an atom is a set $E \in \mathcal{M}$ s.t. $F \in \mathcal{M}, F \subset E$ $\implies \mu(F)=0$ or $\mu(F)=\mu(E)$]. ...
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0answers
41 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
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0answers
18 views

A basic question on measurable rectangles

In Billingsley, the following statement has been written : As $\mathscr{R^1} \times \mathscr{R^1} = \mathscr{R^2}$, $\mathscr{X} \times \mathscr{Y}$ is in general much larger than the class of ...
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0answers
45 views

Mean value formula

Let $u(x)$ be an entire positive solution of the equation $$\Delta u - u = 0 ~~~on ~ R^{n} ~~n>1$$ (a) Can you find a mean value formula? (b) Let $\mu$ be a positive Radon Measure on the unit ...
1
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1answer
28 views

A measurable function equal to a countable sum of characteristic functions?

A theorem in measure theory says that if $\mu$ is a measure on $X$ and $f : X \rightarrow [ 0, \infty]$ is $\mu$-measurable, then there exists a sequence $( A_k)_{k \in \mathbb{N}}$ of ...
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1answer
28 views

Question about $\sigma$-Algebra of measurable function

If I have a $\mathcal{A}-\mathcal{A}'$ measurable function $$T : \Omega \to \Omega' $$ according to measurable spaces $(\Omega ,\mathcal{A})$ and $(\Omega' ,\mathcal{A}')$, and $B\subset ...
0
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1answer
35 views

Radon measure, Integral

Let $d \geq1$ and $D\subset \mathbb{R}^{d}$ be open In the following, $J$ is a symmetric positive radon measure on $D\times D \setminus \operatorname{diag} $, where $\operatorname{diag}=\{(x,x)\mid x ...
3
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0answers
42 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
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0answers
25 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
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1answer
48 views

Haar measure on $G \times G$, where $G$ is compact

Let $G$ be a compact group. Let $\mu'$ and $\mu$ be the Haar measure on $G \times G$ and $G$, respectively, and further such that $\mu'(G \times G) = 1$ and $\mu(G)=1$. Does it follow that $\mu' = \mu ...
0
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1answer
27 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
2
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1answer
33 views

Poincaré Recurrence Theorem (measure theory version)

I had a look on the proof of the following Recurrence Theorem of Poincaré: Let $(\Omega,\Sigma,T,m)$ be a conservative dynamical system in measure theory for which the function $T^{-1}$ ...
2
votes
2answers
38 views

Show that $h(A)=\int_A f \, d\mu$ is a measure.

Let $f:\Omega \rightarrow \mathbb{R}$ be a simple function on a measure space ($\Omega,\Sigma,\mu$). Let $h:\Sigma \rightarrow \mathbb{R}$ a function with $h(A)=\int_A f \, d\mu$. Show that ...
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0answers
32 views

Basic facts related to Haar measure

I have a compact group $G$ and continuous functions $f_1, f_2$ from $G$ to $\mathbb{C}$ and $g: \mathbb{R} \rightarrow \mathbb{C}$. I have two questions related to Haar meausure. Is it true that $$ ...
0
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1answer
31 views

The range of the distributional laplacean, defined in $W_0^{1,1}(\Omega)$.

Let $\Omega\subset \mathbb{R}^N$ be a bounded, smooth domain. Assume that $u\in W_0^{1,1}(\Omega)$ and consider the distributional laplacean of $u$; $\Delta u$. My question is: when is $\Delta u\in ...
2
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2answers
73 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
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1answer
26 views

Showing that a set is in terminal $\sigma$-Algebra

I am reading a probability theory book (from Bauer) and I found the following statement in the book that I cant understand: Given a sequence of independent random variables $(X_i)_{i\in\mathbb{N}}$ ...
4
votes
2answers
44 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
4
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2answers
46 views

Show that f is measurable

Let $U$ be a open Set of $\mathbb{R} \times [0,\infty]$ and let f be defined as $$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$ How can I show that $f$ is ...
1
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1answer
39 views

Disprove counterexample for $\limsup A_n = \emptyset$

Let $(A_n)_{n=1}^{\infty}$ be a pairwise disjoint collection. $\lim A_n = \emptyset$? (see here and there) What about a set of extended real numbers $A_n=(n,n+1]$? It seems that $\limsup A_n = ...
1
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1answer
39 views

Given a pairwise disjoint collection, $\liminf A_n = \emptyset$?!

Let $(A_n)_{n=1}^{\infty}$ be a pairwise disjoint collection. $\liminf A_n = \emptyset$?! Spin-off from here. My attempt: $\liminf A_n$ $= \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} A_n$ $= ...