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Questions tagged [measure-theory]

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

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Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
timofei's user avatar
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22 votes
0 answers
293 views

Measurability with respect to equivalent sub-$\sigma$-algebras for mappings into countably generated spaces

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $\mathcal{G}_1$ and $\mathcal{G}_2$ be sub-$\sigma$-algebras of $\mathcal{F}$ that are equivalent to each other in the sense that $...
Julian Newman's user avatar
22 votes
0 answers
541 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
Dawid C.'s user avatar
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20 votes
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Errata for Dieudonné's Treatise on Analysis volume 2 second edition

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
brunoh's user avatar
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19 votes
0 answers
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Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

This is a verification request for a counterexample that I think I have found for Problem 32.13 on page 427 in Patrick Billingsley’s Probability and Measure textbook (third edition, but the problem ...
triple_sec's user avatar
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19 votes
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885 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
Paul-Benjamin's user avatar
18 votes
0 answers
380 views

Is the power of $2$ in the Euclidean norm related to the fact that the algebraic closure of the reals is $2$-dimensional?

Consider any local field $K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as $$|x| = \frac{\mu(xS)}{\mu(S)},$$ where $\mu$ is any Haar ...
pregunton's user avatar
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17 votes
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400 views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
0xbadf00d's user avatar
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17 votes
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Minimum area contained between measurable set and translate by $\lambda$: A strengthening of 2018 USA TSTST #9

Question Given $\lambda\in\mathbb{R}^+$, what is the smallest possible $c$ for which, given any measurable region $\mathcal{P}$ in the plane with measure $1$, there always exists a vector $\mathbf{v}$...
Carl Schildkraut's user avatar
16 votes
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401 views

Is it true that Lebesgue's differentiation theorem follows from Lebesgue's density theorem?

Let $(X,d)$ be a separable complete metric space and $\mu$ a probability measure on the Borel subsets of $(X,d)$. Suppose that the Lebesgue's density theorem holds, i.e. that for each Borel set $A$ of ...
Bob's user avatar
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15 votes
0 answers
330 views

Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?

Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}^2$ of length $1$, the set $\...
Saúl RM's user avatar
  • 3,535
15 votes
1 answer
1k views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and $\...
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14 votes
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575 views

Is there any condition that makes a measure zero set necessarily countable?

Background : Let us consider the Lebesgue measure space $(\Bbb{R}, \mathcal{L}(\Bbb{R}),m) $. Here measurable set means Lebesgue measurable and measure means Lebesgue measure. $\mathcal{S}\subset \...
Ussesjskskns's user avatar
14 votes
0 answers
1k views

A detailed and self-contained proof of Tonelli's theorem

Motivation: I have seen the interchange of limit/derivative and integral many times, but don't know how such operation makes sense. I've always desired to remove this uncertainty by giving a ...
Akira's user avatar
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14 votes
1 answer
477 views

2-dimensional Lebesgue measure of certain sets in $R^3$

Let $\theta >0$ and $E \subseteq \mathbb{R}^3$ be a closed set (I've added closedness as a new requirement) which satisfies the following condition: For any $x \in E$, there exist at least two ...
Shorty's user avatar
  • 467
14 votes
0 answers
5k views

If $f_n \to f$ and $g_n \to g$ in measure and $\mu$ is finite, then $f_n g_n \to fg$ in measure

This is Problem 3.1.5 in Cohn's Measure Theory, 2nd edition. Let $\mu$ be a measure on $(X, \mathcal A)$, and let $f, f_1,f_2, \ldots$ and $g,g_1,g_2,\ldots$ be real-valued $\mathcal A$-measureable ...
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14 votes
1 answer
1k views

Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
Pantelis Sopasakis's user avatar
13 votes
0 answers
404 views

Any reference on Jensen inequality for measurable convex functions on a Banach space?

The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be ...
P. Quinton's user avatar
  • 6,041
13 votes
0 answers
350 views

Hardy's inequality proof using Doob's inquality

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
mathex's user avatar
  • 642
13 votes
0 answers
293 views

Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
WillG's user avatar
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13 votes
0 answers
328 views

Lebesgue Measure: Is it Optimal?

Consider a collection $\Sigma$ of subsets of $\mathbb{R}^d$ such that $A, B \in \Sigma \implies (A \cup B \in \Sigma \hspace{0.3mm} \text{ and } A \cap B \in \Sigma \hspace{0.3mm} \text{ and }...
Thomas Winckelman's user avatar
13 votes
0 answers
522 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
David C. Ullrich's user avatar
12 votes
0 answers
137 views

"Derivative" of function on set of measures

Across my studies I came across an object which could be thought of as a "derivative" of a function defined on a set of measures and I was looking for references to the matter (if any). ...
Brandon's user avatar
  • 3,185
12 votes
0 answers
359 views

Want to study Graduate Measure Theory with heavy Emphasis on Topology and/or Geometry.

I did one course in Measure Theory and want to study it again. But this time I want to do this in a way that emphasizes Measure Theoretic structure on Geometric or Topological Spaces. I don't know, if ...
Sagnik Biswas's user avatar
12 votes
0 answers
501 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
Pedro Vaz Pimenta's user avatar
11 votes
0 answers
589 views

Understanding Lang's Proof of Fubini's Theorem

This question concerns the proof of Theorem 8.4 (Fubini's Theorem part 1) on page 162 in Lang's real and functional analysis book. To understand the proof I need to give following background from the ...
Alphie's user avatar
  • 4,797
11 votes
0 answers
5k 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}...
Infinitebig's user avatar
11 votes
0 answers
280 views

A characterization of the "Direct Integral" construction in terms of the properties it satisfies?

Fortunately there's a wonderful thing called the "direct integral" which enables one to make sense of direct sums of uncountably infinite families of Hilbert spaces. Unfortunately I've tried to read ...
Saal Hardali's user avatar
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11 votes
0 answers
7k views

Integral over set of measure zero.

Let $f:(X,\eta,\mu)\to [0,\infty]$ measurable and $E\in\eta$ with $\mu(E)=0$ ($\eta$ is a $\sigma$-algebra, $\mu$ a positive measure). Then $\displaystyle\int_E f\,d\mu=0$. We have $$\int_E f\,d\mu=\...
user126033's user avatar
  • 1,044
11 votes
0 answers
795 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
Sinusx's user avatar
  • 625
11 votes
0 answers
6k views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
Zach's user avatar
  • 306
11 votes
0 answers
640 views

Why is the Radon-Nikodym derivative needed for products of complex measures?

Given $\mu, \nu$ complex Borel measures on $\mathbb{R}^n$, then product measure $\mu \times \nu$ on $\mathbb{R}^n \times \mathbb{R}^n$ is defined by $$d(\mu \times \nu)(x, y) = \frac{d\mu}{d|\mu|} (x)...
user43091's user avatar
  • 111
11 votes
0 answers
2k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
precarious's user avatar
10 votes
0 answers
391 views

Radon-Nikodym derivative of SDE law

Let $(\Omega, (\mathcal F, \mathbb P)$ be a probability space, $T$ some index set, and $(S, \Sigma)$ a measurable space. Let $X^f$ and $X^g : T × \Omega \to S$ be stochastic processes defined by the ...
PleaseAnswerMyQuestion's user avatar
10 votes
0 answers
420 views

Stone Duality: What are $\sigma$-Algebras Dual To?

Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of ...
user avatar
10 votes
0 answers
566 views

Practical applications of non-standard probability

Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A)...
alfalfa's user avatar
  • 1,499
10 votes
0 answers
1k views

Are Dynkin's $\pi-\lambda$ Theorem and the Monotone Class Theorem equivalent?

Let $X$ be a set. If $\mathcal{A}$ is a family of subsets of $X$ then let $\sigma(\mathcal{A})$ denote the $\sigma$-algebra generated by $\mathcal{A}$. Definition $1$ A $\pi$-system on $X$ to be a ...
Anguepa's user avatar
  • 3,129
10 votes
0 answers
783 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 ...
EconometricsPerson's user avatar
10 votes
0 answers
239 views

Halmos Measure Theory section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
Sergio's user avatar
  • 3,409
10 votes
0 answers
261 views

Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
Zach466920's user avatar
  • 8,341
10 votes
0 answers
5k views

Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
mrk's user avatar
  • 3,095
9 votes
0 answers
317 views

Haar measures are decomposable

In the real analysis book by Folland, section $11.1$ exercise $9$ have been come that: if $G$ is a locally compact topological group with Haar measure $\mu$, then $\mu$ is decomposable. A measure ...
Amirhossein Haddadian's user avatar
9 votes
0 answers
254 views

Measure that is translation invariant but not invariant to negation

Does a measure $\mu: \mathscr{B}^1\to[0,\infty]$ (where $\mathscr{B}^1$ is the Borel sigma-algebra) that is translation invariant (i.e. $\mu(x)=\mu(c+x)$) but not invariant under $t:\mathbb{R}\to\...
stack_math's user avatar
9 votes
0 answers
150 views

Can I split $E$ in equal volume parts?

Problem: Let $E \subset \mathbb{R}^N$ be a connected, bounded, open and smooth (or just $N$-measurable) set and denote with $\mathcal{L}^N$ the Lebesgue measure. Define $\Omega_i = \{ x \in \mathbb{R}^...
Filippo Giovagnini's user avatar
9 votes
0 answers
352 views

The volume of $k$-parametrised manifolds when $k=1$ or $k=2$ or $k=3$ agrees respectively with the usual intuitive idea of length or area or volume.

James Munkres, at the chapter $22$-th of the text Analysis on Manifolds, gives the following definition. Furthermore, to point out I say that the function $V\left(D\alpha\right)$ corresponds with ...
Antonio Maria Di Mauro's user avatar
9 votes
0 answers
243 views

What is the epistemological status of the usual proof(s) of Pythagoras' theorem?

Pythagoras' theorem has a variety of geometric proofs, such as: I want to teach at least one of these proofs to my high school students, because it shows that the formula $\|(x,y)\| = \sqrt{x^2 + y^2}...
goblin GONE's user avatar
  • 67.9k
9 votes
0 answers
307 views

Is Hilbert's space-filling curve measure preserving?

Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...
Christian Chapman's user avatar
9 votes
0 answers
2k 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 $\cup_n \mathcal F_n$. ...
Gabriel Romon's user avatar
9 votes
0 answers
573 views

Proving $\lim\limits_{n\to \infty}\int\limits_{0}^{1} f_n(x)dx=0$

If $\{f_n\}$ is a sequence of continuous functions on $[0,1]$ such that $0\leq f_n\leq 1$ and such that $f_n(x)\to 0$ as $n\to \infty$, for every $x\in [0,1]$, then $$\lim\limits_{n\to \infty}\int\...
user160110's user avatar
  • 2,710
9 votes
0 answers
199 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
triple_sec's user avatar
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