Questions tagged [measure-theory]

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

8,116 questions with no upvoted or accepted answers
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Time derivative of a measure and convolution

Suppose $\mu_t : [0, + \infty) \to \mathcal{P}(\mathbb{R}^n)$ is a curve of probability measures on $\mathbb{R}^n$. For each $\nu \in \mathcal{P}(\mathbb{R}^n)$ I define $\nu \ast \rho$ as the ...
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74 views

Topology and $\sigma$-algebra on the space of probability measures

Let $(X,\mathbf{A})$ be a $\sigma$-algebra of sets, $\mathscr{M}(\mathbf{A})$ be the set of all $\mathbf{A}$-measurable functions $f\colon X\to [0,1]$, and $\mathscr{P}(\mathbf{A})$ be the set of all ...
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79 views

Lower bound on the integral of averaging operator.

In the following I denote by $B(x,r)$ the ball centered at $x$ of radius $r$, and by $|B(x,r)|$ the measure of this ball. I am trying to solve the following exercise. Question Let $f \geq 0, R \geq ...
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544 views

$f$ is Riemann integrable if the set of discontinuities is measure zero.

Let $f$ be a bounded function on a compact interval $J$, and let $I(c,r)$ denote the open interval centered at $c$ of radius $r>0$. Let $osc(f,c,r)=\sup|f(x)-f(y)|$, where the supremum is taken ...
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197 views

Direct proof that the irrationals on $[0,1]$ have measure $1$

I am seeking a “direct” proof that the Lebesgue measure of the irrationals on the unit interval is $1$. The standard proof I see is that the measure of the unit interval is 1, and the rationals have $...
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1answer
190 views

limsup of continuous function is measurable.

I want to show that if $F$ is continuous on $[a,b]$ then $$\limsup_{h \rightarrow0, h>0} \frac{F(x+h)-F(x)}{h}$$ is measurable. By the definition of $\limsup$ we can write \begin{align} &\...
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118 views

Is there a continuous and injective function $f: [0,1] \to \mathbb R^2$ such that the image has positive two dimensional Lebesgue measure?

Is there a continuous and injective function $f: [0,1] \to \mathbb R^2$ such that the image has positive two dimensional Lebesgue measure? I am not sure how to approach here. I think ine way to ...
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78 views

Continuous function $f:\mathbb R\to \mathbb R$ such that $f(B)$ is not measurable for a Borel set $B$.

1) I saw in a book that there are $\mathcal C^1(\mathbb R)$ function $f:\mathbb R\to \mathbb R$ such that $f(B)$ is not measurable for $B$ a Borel set. I don't really find such an example. Any idea ? ...
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119 views

Applications of Sard's theorem?

I'm writting an essay for University about differential topology. In particular, I'm studying the applications of Sard's theorem on differential topology. I have included the proof of Whitney's ...
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147 views

How to use the Lindeberg CLT in this scenario ? (analysis problem)

We have $(X_i)_{i \in \mathbb Z}$ iid random variables with $1\le X_i \le2$ almost surely. We define $X(x,\omega) \equiv X_i (\omega)$ if $x\in [i,i+1[$ and $X_\epsilon (x, \omega) \equiv X(x/\...
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230 views

Show measure zero: application of Theorem 1 in Lang (1986)?

Suppose I have a random variable $Y$ with support $\{1,2,..., M\}$. Consider a random vector $V\equiv (V_1, V_2,..., V_M)$ with support $\mathcal{V}\subseteq \mathbb{R}^M$ with positive Lebesgue ...
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394 views

Convergence in probability of sequence of independent random variables implies almost sure convergence of their sum?

$\{X_n\}^{\infty}_{n=1}$ are independent random variables. Let $S_n = X_1 + \cdots + X_n$ be the random walk. Show that $\{S_n\}$ converges almost surely if and only if the random sequence converges ...
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1answer
188 views

The $\mathbf{F}$-metric induces the weak topology on the set of bounded varifolds

Some preliminary definitions and notation: (1) Given a vector space $\mathbb{V}$, we denote by $G_k(\mathbb{V})$ the $k$-grassmannian of $\mathbb{V}$, i.e. the set of all $k$-dimensional vector ...
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91 views

Underlying Random Variable of Conditional Expectation

Consider interval $[0,1]$ with its Borel $\sigma$-algebra and Lebesgue measure on it. It is known that $f$ is an integrable function on $[0,1]$. $\mathcal{F_n}=\sigma([\frac{k-1}{2^n},\frac{k}{2^n}))$ ...
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169 views

Proving the Fubini-Tonelli Theorem

Okay, so the notes I'm using give the theorem as follows: Let $ (\Omega_1, \mathcal{A}_1, \mu_1) $ and $ (\Omega_2, \mathcal{A}_2, \mu_2) $ be two $ \sigma-$finite measure spaces. $ (1) $ If $ f \...
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2answers
243 views

Absolute continuity of a Borel measure

This is a question from Ph. D Qualifying Exam of real analysis. Let $F$ be an increasing function on $[0,1]$ with $F(0)=0$ and $F(1)=1$. Let $\mu$ be a Borel measure defined by $\mu((a, b))=F(b-)-F(a+...
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142 views

If the solution is too simple, it must be incorrect?

Prove that to each $\epsilon > 0$ there exists a $\delta > 0$ such that $\displaystyle \int_E |f| d \mu < \epsilon$ whenever $\mu (E) < \delta$, where $f \in L^1(\mu)$. I found this ...
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144 views

A density proof of the Dominated Convergence theorem

Consider the following corollary of the DCT, sometimes called the Bounded Convergence theorem: If $f_n$ is a uniformly bounded sequence of functions converging pointwise a.e. to $f$ on a finite ...
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1answer
204 views

Integration by parts and non-absolutely continuous distributions

Let $x\in [a,b]$ be a real random variable with distribution $H$ that is not absolutely continuous (w.r.t Lebesgue measure). I saw this in a paper: $$ \int_a^b xH(dx) = b-\int_a^bH(x)dx. $$ I get it ...
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1k views

Fourier transform of $L^1$ function is continuous function

I would like to know if my proof is correct. Define Fourier transform of $f\in L^1(\mathbb{R}^n)$ as $$\widehat{f}(x)=\int_{\mathbb{R}^n}e^{-ix\cdot y}f(y)\text{d}y.$$ I want to show that $\widehat{f}...
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786 views

Intuition regarding the $\sigma$ algebra of the past (stopping times)

Let $(\mathcal F_n)$ be a filtration and $\tau$ be a stopping time with values in $\mathbb N \cup \{\infty\}$. Let $\mathcal F_\infty$ be the sigma-algebra generated by $\cup_n \mathcal F_n$. ...
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1answer
106 views

When is a measure the pushforward of another measure?

Let $(X,\Sigma)$ be a measurable space. Let $\mu$ and $\nu$ be two measures on thereon. Are there any reasonable restrictions on these measures to ensure the existence of a (measurable) function, $f: ...
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337 views

Is an essentially bounded continuous function bounded?

I've just started working with $L^p$ spaces, and I've learned that a function $u:\mathbb{R}^n \rightarrow \mathbb{R}$ is essentially bounded if there exists a constant M such that $\{x \in \mathbb{R}^...
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1answer
83 views

Showing that every finite measure on $(X,P(X))$ can be written as $\int_E f \,du$.

I want to show that every finite measure on $(X,P(X))$, where $P(X)$ denotes the power set of $X$ has the form $v(E) = \int_E f \,du$ for a non-negative measurable function $f$ and where $u$ is the ...
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69 views

When do we have $\sup_{x}\int fdy=\int \sup_{x}fdy$?

When do we have $$\sup_{x}\int fdy=\int (\sup_{x}f)dy$$? Here, $f$ is a function of $x$ and $y$. Some say: Use The monotone convergence theorem, but I really don't understand that hint.
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309 views

The Borel $\sigma$-algebra on $[0,1]$

Let $\mathscr{B}$ be the usual Borel $\sigma$-algebra on $[0,1]$, i.e., the smallest $\sigma$-algebra contained in $\mathcal{P}([0,1])$ which includes the intervals $O \cap [0,1]$, with $O$ open in $\...
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64 views

About the sigma additivity of volume of half-open cubes in $\mathbb{R}^n$

Many texts (Fremlin, Stroock,etc.) on measure theory seem to think this is a difficult lemma to prove. Tao proves this in a indirect way by approximating volume with the number of grid points. If $...
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192 views

Topological structure & compactness of the space of probability measures

Let $(X,\tau)$ be a topological space and let $\mathcal P$ denote the set of all probability measures on $\mathscr B$, the Borel $\sigma$-algebra generated by the open subsets of $X$. In what follows, ...
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211 views

Non-probabilistic analogue of the Second Borel-Cantelli lemma

The first Borel-Cantelli Lemma says that if we have events $E_i$ and $\sum_iP(E_i)<\infty$ then the probability infinitely many events occur is 0. The second is a partial converse saying if the ...
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2answers
113 views

Showing $f\ge1$ a.e on $[a,b]$

Let $f$ be $\in L[a,b]$ Assume for any subinterval $I \subset [a,b]$ we have $\int_I f \geq |I|$ show that $f \geq 1$ a.e on $[a,b]$. I started with a proof by contradiction. Assume there exists a ...
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105 views

If $xf \in L^2(\Bbb R)$, is $f \in L^1(\Bbb R)$?

I tried using Holder's inequality with $|f| = |f| \frac{(1+|x|)^{\frac{1}{2}}}{(1+|x|)^{\frac{1}{2}}}$, or variant but I can't seem to make it work. Any help is appreciated.
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426 views

Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 33.9(c)

This is yet another verification request on a(n allegedly) false statement in Patrick Billingsley’s Probability and Measure textbook. I have been self-studying this excellent book for quite a while ...
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90 views

A question on a proof of the existence of non-measurable sets

This is an excerpt of Rosenthal's book: "A First look at Rigorous Probability Theory" and contains an important theorem in the development of measure theory. While I generally understand the proof ...
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78 views

Measurability of $f\otimes g:(X\times Y,\mathcal{B}(X)\otimes \mathcal{B}(Y)) \to (H_1 \otimes H_2,\mathcal{B}(H_1\otimes H_2))$

Assume that $(X,d)$ and $(Y,\rho)$ are metric spaces, and that $f:X \to H_1$ and $g:Y\to H_2$ are isometries (especially Borel measurable) into Hilbert spaces. Now define the mapping $$ f\otimes g:X\...
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269 views

Is it true that every finite sigma algebra is 'isomorphic' to some power set sigma algebra?

Let $(\Omega_1,\mathcal {F_1})$ and $(\Omega_2,\mathcal {F_2})$ be two measurable spaces.we call $\mathcal{F_1}$ and $\mathcal {F_2}$ are isomorphic if there exist a measurable map $f: (\Omega_1,\...
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436 views

Is there a unique finitely additive extension of a countably additive product probability measure to the product $\sigma$-algebra for finite products?

Consider a finite collection of measure spaces $\{(A_i, \mathcal{A}_i)\}_{i=1}^N$ where each $\mathcal{A}_i$ is a $\sigma$-algebra. Let $\mathcal{A}$ be the algebra on $\prod_{i=1}^N A_i$ generated by ...
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1answer
185 views

Alternative definition of a submartingale, problem with the Radon-Nikodym theorem.

Assume you have a stochastic base $(\Omega, \mathcal{F},P,\mathbb{F})$. A submartingale is usually defined as an adapted process for each $t$ $E(|X_t|)<\infty$ and $E(X_t|\mathcal{F}_s)...
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314 views

Conditional expectation continuous in the conditioning argument?

Let $X$ and $Y$ be random vectors defined on a common probability space. $X$ takes values in a finite-dimensional space $\mathcal{X} \subset \mathbb{R}^p$, while $Y$ takes values in $\mathbb{R}$. The ...
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391 views

Cantor sets of positive measure

The perfect set property theorem states that every uncountable Borel set contains a compact subset homeomorphic to Cantor set. Now suppose that $\mu$ is a regular Borel measure (on some measurable ...
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129 views

Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
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229 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times X$...
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164 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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220 views

Example of an oscillation Young measure

I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get ...
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112 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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339 views

Defining a Sum of Random Variables

Given that a (real) random variable $X$ is a measurable map from a probability space $(\Omega,\mathcal{A},P)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, how do we define a sum of random variables $X+Y$...
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155 views

Help to understand a Lemma about 'supremum of a family of measures'

I read the following Lemma from a paper but I can't understand the proof. Please help! Lemma: Let $\mu$ be a positive measure defined on the family of open subsets of $\Omega$, which is super-...
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1answer
307 views

Product of $L^{1}$ Function and Exponentially Integrable Function

Problem. Let $g\geq 0$ be in $L^{1}[0,1]$ and suppose that $\int gfdx<A$ whenever $\int e^{f}dx\leq 1$. What can one say about $|\{g>\lambda\}|$ for $\lambda\gg 1$. Is $g\in L^{2}[0,1]$...
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108 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min \...
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123 views

Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.

Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
5
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1answer
552 views

Absolutely continuous function on R

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole $...