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

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12 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$

Could you please help me solving this old prelim problem. Any hints are appreciated
2
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0answers
12 views

Boolean algebra with measures

Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic ...
7
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1answer
64 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
7
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2answers
32 views

Is every measurable set a measure-independent limit of open sets

My main question is Q1. Let $B$ be a Borel-measurable subset of $\mathbb R$. Is there a sequence of open sets $U_n$ independent of any measure such that for all Borel probability measures ...
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1answer
48 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
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1answer
22 views

Measurability of adapted processes

Let $(\Omega, \mathscr{A}, P)$ be a probability space, $(E, \mathscr{E})$ a measurable space and $X_t : \Omega \to E$, $t \geq 0$ a measurable stochastic process, i.e. the map $X : [0, \infty) \times ...
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0answers
13 views

For $E \subset \mathbb{R}$ and $\epsilon >0$, $\exists$ $(a,b)$ s.t. $\theta(E \cap (a,b)) \geq (1-\epsilon)|b-a|$ ($\theta$ Lesbegue Outer Measure)

In my notes this statement is left unproven. I want to show that for any measurable set $E \subset \mathbb{R}$ with $\theta(E)>0$, there exists an interval $(a,b)$ that covers $E$ arbitrarily ...
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0answers
53 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
0
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1answer
33 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
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0answers
21 views

a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
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1answer
36 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
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2answers
43 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
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2answers
30 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
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2answers
49 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
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1answer
37 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
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1answer
26 views

Showing convergence of a series almost everywhere

If $\sum_{k=1}^\infty a_k$ is convergent series of positive terms and $(\alpha_k)_{k\in \Bbb N}$ is a sequence of real numbers, then the series $$\sum_{k=1}^\infty\frac{a_k}{\sqrt{|x-\alpha_k|}}$$ ...
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2answers
29 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...
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0answers
34 views

How do I see that $|f_n -f|^p = \lim_{n \rightarrow \infty} \inf |f_n - f_{n_k}|$? [on hold]

Let $(X, \mathcal E, \mu)$ be a measure space, $p \in [1, \infty]$ and $f, f_n \in \mathcal M(\mathcal E)$ Suppose $(f_n)$ is Cauchy in $\mu$-p-mean and $f_{n_k} \rightarrow f$ converge ...
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0answers
20 views

Problem with the definition of semi-ring and $\sigma$-sets

I have a problem with a statement I found concerning the definition of semi-ring and that of $\sigma$-set. So, here there is. Assume the definition of a semi-ring $\mathcal{S}$ over a non-empty set ...
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0answers
15 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
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0answers
14 views

Probability measure of rank-$r$ matrices

I have a question about the distribution of matrices with a specific rank. Consider $\mathcal{M}^{n\times m}$ the set of all $n \times m$ matrices with entries in some field $\mathbb{K}$. If I define ...
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2answers
39 views

Minkowski's Inequality in $L^\infty$ space

How can one show the inequality that $\|f+g\|_\infty ≤ \|f\|_\infty + \|g\|_\infty$? Can we use same real number $a$ for both $f$ and $g$ ? i.e, $$\|f\|_\infty = \text{ess} ...
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0answers
15 views

Cinlar Ex. 1.15: Trace space of a measurable space.

In constructing the trace space on a subset of a measurable space, it seems one has to assume that the subset is an element of the original measure space's sigma algebra, i.e., measurable in the ...
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2answers
65 views

Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
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1answer
16 views

Locally compact metric space, Urysohn, approximation

Let $E$ be a locally compact separable metric space, $\mathcal{B}(E)$ be the $\sigma$-algebra of $E$ and $m$ be a $\sigma$-finite borel measure on $(E,\mathcal{B}(E))$. Assumtion There exists a ...
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0answers
34 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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2answers
35 views

prove that $F$ is $\mu\times\mathcal{L}$ measurable where $F(n,x)=\frac{(2n+1)^2\sin((2n+1)x)}{(n(n+1))^2}$

Let $\mu$ be the counting measure on $\mathbb{N}$ and $\mathcal{L}$ be the Lebesgue measure on $[0,\pi]$. Define the function $F$ on $\mathbb{N}\times\mathcal{L}$ by ...
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3answers
37 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
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2answers
24 views

Looking for proof of theorem on complex measurable functions

In University I have been given the following result: If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set ...
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2answers
58 views

Prove that a intergral over $\mathbb R$ is finite

Let $K\in \mathcal L_1(\mathbb R)$ and $f$ be measurable and bounded on $\mathbb R$ such that $\lim_{|x|\to \infty} f(x)=0$. Define $$F(x):= \int _{\mathbb R} K(x-s)f(s)\;ds \qquad (x\in \mathbb R)$$ ...
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1answer
24 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
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0answers
53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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0answers
35 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
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1answer
19 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
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1answer
28 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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1answer
16 views

A relation between the inner and outer jordan measures

I'm studying measure theory and I was thinking about the following question: Is it true that whenever $A\subset B\subset \mathbb{R}^n$ are bounded, $$m^*(B-A)=m^*(B)-m_*(A)?$$ I have easily ...
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1answer
17 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
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0answers
13 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform ...
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1answer
35 views

Measurable set limit

If $\forall n \in ℕ$ , $ f_n: (X,M) \rightarrow (\overline{\mathbb{R}},B) $ are measurable. (where X is any space, M is a sigma algebra, B is Borel sigma algebra) Prove that the set $A = \{x\in X: ...
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1answer
26 views

How to show that the function $g(x)=x|\sin(x^{-1/2})|$ is absolutely continuous?

I am having trouble showing the on $[0,1]$, $g(x):=x\mid\sin(x^{-1/2})\ \mid$ when $x>0$ and $0$ is $x=0$ is absolutely continuous. I was instructed to try: $\ m(A) < \delta \Rightarrow ...
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5answers
120 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
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1answer
30 views

Relation between the modulus of integrability and $L^p$ spaces

Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Given an integrable function $f$ on $X$, we can quantify its integrability in multiple ways. One is the modulus of integrability, which is a ...
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2answers
287 views

Measure - exercise 22 from Folland

I'm doing some exercises from Folland's real analysis book. Exercise 18 is done and should help to do exercise 22, but I'm stuck. The definition of completion is given below. This is not ...
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0answers
42 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
3
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1answer
17 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
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0answers
51 views

Why does the union of all open null sets is itself a nullset for second countable space?

On the online Encyclopedia of mathematics, it is written "The existence of a countable base guarantees that the union of all open μ-null sets is itself a nullset." See: ...
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0answers
27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
1
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0answers
33 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
1
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2answers
54 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
4
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2answers
49 views

If $\int_E f=\int_E g$ then $f=g$ a.e.?

Is the converse of the following statement is true? Let $f$ and $g$ be two bounded measurable functions on a set $E$. If $f(x)=g(x)$ a.e. on $E$ then $$\int_E f=\int_E g$$ Here is my proof for ...