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

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Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...
2
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1answer
49 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
2
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1answer
41 views

When is the image of a $\sigma$-algebra a $\sigma$-algebra?

Let $(E,\mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces and $f:E \rightarrow F$ with $f$ $\mathcal{E}/\mathcal{F}$ measureable. When is $f(\mathcal{E})$ a $\sigma$-algebra? I am aware that ...
0
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0answers
28 views

General step functions for lebesgue integral

For simplicity, I will only assume we are talking about the lebesgue integral on the same line. I read a construction of the riemann integral, that was designed in a way to resemble the construction ...
7
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1answer
108 views

Is there any $F \in \mathscr{F}$ such that $\mu(F)=x$?

Let $ (\Omega,\mathscr{F},\mu)$ be a probability space such that $\mu$ is non-atomic, and fix $x \in [0,1]$. Is it true that one can find $F \in \mathscr{F}$ for which $\mu(F)=x$? And what if $\mu$ is ...
0
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1answer
26 views

Show that $\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$ is a $\sigma$-algebra

Let $\{A_{i}\}_{i = 1}^{n}$ be a family of pairwise disjoint subsets of $X$. It is said that $$\mathcal{F}:=\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$$ is a $\sigma$-algebra. ...
3
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2answers
53 views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, ...
4
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1answer
21 views

Weak $L^p$ implies strong $L^q$ for $q<p$

Another prelim question... Suppose $0<q<p<\infty$, and $E\subseteq \mathbb{R}^n$ has finite measure. Suppose $f$ is in weak $L^p$, i.e. $\lambda(|f| > t) \leq N/t^p$. Show $f \in L^q(E)$ ...
2
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1answer
57 views

On the space $L^0$ and $\lim_{p \to 0} \|f\|_p$

For $0 < p < \infty$, the definitions of the spaces $L^p$ are very natural. Then, we of course want $L^\infty$ and $L^0$ to be some kind of limits of $L^p$ spaces. What does the parameter $p$ ...
2
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0answers
30 views

Separating family of functions for measures

Given a family $X$ of real-valued functions $f\colon\Omega\to\mathbb{R}$ on some set $\Omega$, consider the smallest $\sigma$-algebra on $\Omega$ making all $f\in X$ measurable with respect to the ...
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1answer
42 views

A problem about Hardy-Littlewood maximal function from Folland real analysis book

For $x \in \Bbb R^n$, define $H^* f(x)=\sup{\frac{1}{m(B)}\int_B|f(y)|\,dy}$, where $B$ is a ball and $x \in B$ and $$H f(x)=\sup_{r>0} \frac{1}{m(B(x,r))} \int_{B(x,r)}|f(y)| \, dy.$$ Show that ...
4
votes
2answers
230 views

Is “random variable” really random?

This is a concept question. The fundamental of modern probability theory is measure theory. A probability space is just a finite measure space and a random variable is just a measurable function. We ...
4
votes
2answers
84 views

If $\sup_n\int_E f_n(x)\ \mathsf dx\leq M\mu(E)$ then the measure of $\{x\in [0,\infty)\mid f(x)>M\}=0$.

This question came up when I was studying for an analysis qualifying exam: Suppose $f_n\geq 0$ for all $n\geq 1$, $f_n\rightarrow f$ a.e. on $[0,\infty)$ and there exists $M>0$ such that ...
3
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1answer
52 views

Measure spaces proof

This theorem comes from the book Real Analysis by Folland Note: $M$ is a $\sigma$-algebra Suppose that $(X,M,\mu)$ is a measure space. Let $\mathcal{N} = \{N\in M: \mu(N) = 0\}$ and $\bar{M} = ...
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0answers
23 views

Let $g(x)=f(x)+x$, where $f(x)$ is the Cantor function from $[0,1]$ to $[0,1]$. prove that $B$ is Lebesgue measurable but not Borel measurable.

Let $g(x)=f(x)+x$, where $f(x)$ is the Cantor function from $[0,1]$ to $[0,1]$. We know for the Cantor set $C$, $g(C)$ contains a nonmeasurable set A. Let $B=g^{-1}(A)$, prove that $B$ is Lebesgue ...
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1answer
25 views

Klenke's definition on exchangeable families of random variables

In the beginning of Ch12 (page 231) of Klenke's book "Probability theory", I find his definition on exchangeable families of random variables confusing. Let $I$ be an arbitrary index set and let ...
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2answers
35 views

Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.

Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space, and suppose each set in $\mathcal A$ has strictly positive measure. Show that $\mathcal A$ is at most ...
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2answers
45 views

Finiteness In Tonelli's And Fubini's Theorems

Why do we suppose in Tonelli's Theorem that $(X,Σ_X , μ_x)$ and $(Y,Σ_Y , μ_y)$ to be σ-finite? Does the theorem fail when it is not? If yes, could someone provide an example? We also suppose in ...
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1answer
47 views

What space is the set of all CDFs?

I'm relatively new to functional analysis and am trying to make comparisons across different CDFs (cumulative density functions), i.e. right-continuous, weakly increasing functions ...
0
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1answer
28 views

Riemann sums of improper integral

I wonder about the following: let $f:[0,\infty]\to\mathbb{R}$ be Lebesgue integrable,i.e. $\int_0^\infty |f(x)|d\lambda(x)<\infty$ Does it hold that $$\lim_{n\to\infty}\int_0^\infty ...
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2answers
55 views

Difference between Null set and empty set

One of my friend asked this doubt.Even in lower class we use both as synonyms,he says that these two concepts have difference.Empty set $\{ \}$ is a set which does not contain any elements,while null ...
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1answer
14 views

Exponential and convergence in $L^2$ bis

This question is a continuation of my question "Exponential and convergence in $L^2$" posted above: Let $(f_k)$ be a sequence of elements of $L^\infty(\Omega)$, which converge in $L^2(\Omega)$ to ...
2
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0answers
26 views

Linear combination of i.i.d random variables

We say that a random variable $X$ satisfies the $(\alpha,\beta)-$condition for some $\alpha>0$ and $\beta>0$ if $$\mathbb{P}(|X|<t)<\alpha t\text{ and }\mathbb{P}(|X|>t)<e^{-\beta ...
4
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1answer
31 views

Help proving generalized Jensen's inequality $\mathbf{E}[f(\cdot,X(\cdot))\mid \mathscr{G}] \geq f(\cdot,\mathbf{E}[X\mid\mathscr{G}](\cdot))$

I'm reading Meyer's seminal work Probability and Potentials (1966), in which he states the following "borrowed" theorem from Dubins "Rises and Upcrossings of Nonnegative Martingales" (1961). ...
6
votes
3answers
56 views

Non-constant $L^1$ function has a non-zero integral over some interval

Prove that if $f\in L^1([0,1],\lambda)$ is not constant almost everywhere then there exists an interval so that $\int_I\!f\,\mathrm{d}\lambda\neq 0$. Here $\lambda$ is the Lebesgue measure. ...
3
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2answers
65 views

Measures, as a supreme, proof

Let $\Sigma$ be a $σ$-algebra over a set $X$, and $μ_1$ and $μ_2$ finite measures in it. It can be shown that the function $μ:\Sigma \to [0,\infty ]$ defined by $$E\mapsto ...
4
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1answer
35 views

Shorter proof of measurability of the set where two measurable functions differ

Let $f,g$ be measurable functions from $\Omega$ into $[0,\infty]$. I want to show that the set where the two functions differ is measurable. i.e. the set $K = \{x\in \Omega: f(x) \neq g(x) \}$ is ...
0
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1answer
32 views

Why is $|\sigma(\{A_1,A_2,\dots,A_N\})|\leq 2^{2^N}$?

I am trying to understand mathematically why $|\sigma(\mathcal{M})|\leq 2^{2^{N}}$ where $\mathcal{M}=\{A_{1},A_{2},\dots,A_{N}\}$ is a finite system of subsets of $X$. I found this below Definition ...
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2answers
51 views

Prove that $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx=a$ for all $a>0$. [duplicate]

This is a qualifying exam problem. Suppose $F$ is a distribution function of a Borel measure $\mu$ with $\mu (\Bbb R)=1$. Prove that $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx=a$ for all $a>0$.
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3answers
76 views

If $\int_A f d\mu =\int_A g d\mu$ and $\mu$ is $\sigma$-finite then $f=g$ a.e [duplicate]

Functions $f,g$ are nonnegative on $(X,\mathcal A, \mu)$. If $\int_A f d\mu =\int_A g d\mu$ for $\forall A\in \mathcal A$ and $\mu$ is $\sigma$-finite then $f=g$ a.e I have no ideas how to prove ...
3
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1answer
37 views

If $\int_A f d\mu \leq \int_A g d\mu$ for all $A$ and $\int_A f$ is $\sigma-$ finite then $f\leq g$ a.e

$f, g$ are nonnegative functions on $(X,\mathcal A, \mu)$ and $\int_A fd\mu \leq \int_A g d \mu, \forall A \in \mathcal A$. Show that if them measure $\nu(A) = \int_A fd\mu$ is $\sigma-$ finite ...
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1answer
15 views

A question on measure from Krengel's book on Ergodic Theorems.

Something which I am not sure how is it inferred. On page 307 of the book Ergodic Theorems by Ulrich Krengel, they write that: " $M_l(x) = \mu(\{ z : k_1(x,z)\geq 1/l \} )$. Let $l(x)$ be the ...
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0answers
14 views

Upper semicontinuity of the function given

Please help me with the following problem. Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be bounded and measurable function. Let $\nu$ be a probability measure. Consider ...
1
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1answer
23 views

Help with Rudin's proof of Riesz Representation Theorem

I am having difficulty understanding a step in the proof of Riesz Representation Theorem, in Rudin's 'Real and Complex Analysis' (P.40, Theorem 2.14): Let $X$ be a locally compact Hausdorff space, ...
0
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1answer
11 views

$S[a,b]$ is dense in $PC[a,b]$ in the $\| \cdot \|_{\infty}$.

I want to prove that $S[a,b]$ (the space of all step functions for all possible finite partitions with the $\| \cdot \|_{\infty}$-norm ) is dense in $PC[a,b]$ (the space of bounded piecewise ...
4
votes
2answers
110 views

Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.

How can we prove the statement? $f$ is Lebesgue integrable if and only if $|f|$ is Lebesgue integrable. First, suppose $f$ is Lebesgue integrable, by definition, $f$ is measurable and nonnegative, ...
7
votes
3answers
62 views

Radon-Nikodym derivative of sum of two measures

Problem Statement: Suppose that $\mu$ and $\nu$ are two finite measures such that $\nu \ll \mu$, let $\rho = \mu + \nu$, and note that since $\mu(A) \le \rho(A)$, and $\nu(A) \le \rho(A)$, we have ...
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1answer
39 views

Exponential and convergence in $L^2$

Let $(f_k)$ be a sequence of elements of $L^\infty(\Omega)$, which converge in $L^2(\Omega)$ to $f\in L^2(\Omega)$. Where $\Omega $ is an open bounded subset of $R^n$. Is it true that : $e^{f_k} $ ...
3
votes
1answer
19 views

Equivalent, finite measures if and only if strictly positive Radon-Nikodym derivative exists

Problem statement: Let $(X,\mathcal{A})$ be a measurable space, and let $\mu$ and $\nu$ be two finite measures. We say $\mu$ and $\nu$ are equivalent measures if $\mu \ll \nu$ and $\nu \ll \mu$ (if ...
1
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1answer
35 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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votes
1answer
51 views

How to write it in words about sigma algebra

(Weak) mathematically: Let $A,B\subseteq X$. If $\sigma(\{A,B\})$ is given and that $A$ and $C$ are atoms in $\sigma(\{A,B\})$, then $\sigma(\{A,B\})=\sigma(\{A,C\})$. Example: Let $X=[0,1]$ and ...
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1answer
45 views

A measure theory book with lots of examples

I find that when learning more abstract concepts, it helps to have a 'simple' example tied to every theorem in order to fully appreciate the theorem or property. However, the course notes I am ...
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1answer
46 views

Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. Prove that $f_n$ converges to $f$ in $L^1$ norm.

Let $\{f_n\}$ and $f$ be Lebesgue measurable functions on $E$ where $|E|<\infty$. Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. (a) Prove that ...
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1answer
40 views

Compact set and measure theory [closed]

I can´t solve this exercises Let $\lambda$ Lebesgue measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $K$ compact set in $\mathbb{R}$ such that $\lambda(K)>0$. For every integer $n \geq 1$ it ...
0
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1answer
20 views

absolute continuous function

If I know that a function $f:R\to R$ is absolute continuous then I know that is has derivative a.e my question is if I look in the limit : $$\lim_{n\to \infty} \ n \cdot ...
0
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1answer
35 views

Baire $\sigma$-algebra

Let $(X,T)$ hausdorff topological space. On the set $X$ it defines the baire $\sigma$-algebra, is written $B_0(X)=\sigma\{f; f \in C(X,\mathbb{R})\}$. Consider now the $\sigma$-algebra ...
0
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1answer
30 views

Another exercises of measure theory

Here another exercises of measure theory: Let $(E,d)$ metric space, and $\mathbb{P}$ probability measure over $(E,B(E))$ . Prove that $\mathbb{P}$ is regular, i.e. if $A \in B(E)$ and $\epsilon>0$ ...
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3answers
47 views

About the cardinality of sigma algebra and power set

Let $X$ be a set. We know that $|\mathcal{P}(X)|=2^{|X|}$. Let $\mathcal{K}=\{A_{1},\dots,A_{n}\}$, where $A_{1},\dots,A_{n}$ are subsets of $X$. Question: Is it true that ...
4
votes
2answers
69 views

Proving a set is Lebesgue Measurable [duplicate]

Measure is a serious weak point of mine, and I cannot figure out this problem: Let $E \subset \mathbb{R}$ be Lebesgue measurable. Suppose that for all open intervals $I$, we have $m(E\cap I) \leq ...
1
vote
1answer
43 views

Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Let $n \geqslant 1$, Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$, where $u \in (0,1)$. What I did is just use the definition of ...