Questions tagged [measure-theory]

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

7,906 questions with no upvoted or accepted answers
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167
votes
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8k views

Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
22
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1answer
2k views

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 ...
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25k views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
19
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0answers
410 views

Explicit description of small open set containing the rationals

We know that the set $\mathbb{Q}$ of rational numbers has measure zero because it is countable. In fact, if $(q_n)_{n=1,2,\ldots}$ is an enumeration of $\mathbb{Q}$, then $\bigcup_{n=1}^\infty(q_n-2^{-...
18
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426 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 ...
17
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1k views

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 ...
15
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186 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 $...
14
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4k views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; i\...
14
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697 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 ...
13
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176 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 ...
13
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1answer
945 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 $\...
12
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217 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 }...
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304 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^...
12
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280 views

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 $\mathbf{v}$...
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305 views

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 ...
12
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1answer
622 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 ...
11
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0answers
406 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 ...
11
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640 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
11
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501 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)...
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1k 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 ...
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147 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 ...
10
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352 views

Definitions of Measurability: Outer-inner measure convergence vs. Caratheodory criterion

If we are looking over subsets of $\mathbb R$ and considering the outer measure defined exactly as $$\mu^*(A) = \inf\left\{ \sum_{k=1}^\infty \ell(I_k) \text{ where the $I_k$ are open intervals such ...
10
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1answer
228 views

Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
10
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214 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 ...
10
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208 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 ...
10
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1answer
870 views

Relation between Shannon Entropy and Total Variation distance

Let $p_1(\cdot), p_2(\cdot)$ be two discrete distributions on $\mathbb{Z}.$ Total variation distance is defined as $d_{TV}(p_1,p_2)= \frac{1}{2} \displaystyle \sum_{k \in \mathbb{Z}}|p_1(k)-p_2(k)|$ ...
10
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1answer
492 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall x \...
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581 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. ...
10
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508 views

Carlson's model and Sierpinski sets

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by ...
10
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0answers
2k views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
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164 views

Defining an unusual subspace of $c_0$

This is going to be a long post, so I'm giving a description first: I recently came across the following exercise: Let $(X,\mathcal{A},\mu)$ be a measure space. If $(f_n)\subset L^p(\mu)$ and $f\in L^...
9
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0answers
349 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)...
9
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0answers
124 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 ...
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0answers
2k views

On the Lebesgue measure of a cartesian product

If $X \subseteq \mathbb{R}^{l}$ and $Y \subseteq \mathbb{R}^{r}$ with $l + r = n$, is it true that $\lambda_{n}(X \times Y) = \lambda_{l}(X) \cdot \lambda_{r}(Y)$ (where $\lambda_{m}$ is the Lebesgue ...
9
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1answer
1k views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and $...
9
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1answer
305 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\...
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169 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 ...
8
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178 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 ...
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291 views

What are “shapes” and/or the polynomials associated to them really called?

While trying to formalize basic Year 8 geometry, I came up with the following: Definition 0. A shape is a CW-complex $S$ together with a topological embedding $f_S:S \rightarrow \mathbb{R}^n$ such ...
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588 views

“Lebesgue” measurabillity on Riemannian manifolds

Let $X$ be a smooth oriented manifold of positive dimension; Let $g_1,g_2$ be two Riemannian metrics on $X$. Define $\Lambda_1,\Lambda_2:C_c(X) \to \mathbb{R}$ by $$ \Lambda_i(f)=\int_X f \, Vol_{...
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188 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
8
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1answer
196 views

Proof that if $f$ is integrable and $\int_E {f}\,{d\mu} = 0$ for all $E\in\Sigma$ then $f=0$ almost everywhere

Proof that if $f$ is integrable and $\int_E {f}\,{d\mu} = 0$ for all $E\in\Sigma$ then $f=0$ almost everywhere My attempt: I called: $A = \{ x\in X : f(x) \ne 0 \}$ $B_n = \{ x\in X: f(x) \gt \...
8
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0answers
328 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 ...
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0answers
217 views

Can an arbitrary probability space be simulated by coin tosses?

Let $B=\left(\{\mathrm{heads},\mathrm{tails}\},\mathcal{P}\{\mathrm{heads},\mathrm{tails}\},\mu\right)$ be the probability space for a single fair coin toss. For any cardinal $\aleph$ let $B^\aleph$ ...
8
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378 views

Clarification on product space, conditional probability measures

Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ ...
8
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0answers
2k 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 ...
8
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969 views

Show that the union over a collection of compact cubes in $\mathbb{R}^n$ is Lebesgue measurable

Let $\mathcal{K}$ be a (not necessarily countable) collection of compact cubes in $\mathbb{R}^n$. Show that $\cup\{K:K\in \mathcal{K}\}$ is a Lebesgue set (Measurable with respect to the Lebesgue ...
8
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0answers
446 views

What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
8
votes
2answers
2k views

Outer Measure of the complement of a Vitali Set in [0,1] equal to 1

I am trying to prove the first part of exercise 33, ch. 1 in Stein and Shakarchi (Real Analysis). I am running into some difficulties following the hint though. Here is the problem (note, $N$ is a ...
8
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1answer
867 views

Lebesgue point of density on $[0,1]$ and Dynkin's theorem

The problem defines a density point $x\in[0,1]$ for a Borel set $A\subset [0,1]$ if $$ \lim_{\varepsilon \rightarrow 0^+} \frac{\mu([x-\varepsilon,x+\varepsilon]\cap A)}{2\varepsilon}=1.$$Denote all ...

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