Questions about abstract measure and integral theory. Also concerns such properties as measurability of maps and sets.

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8
votes
3answers
300 views

reference request : constructive measure theory

As the title said, I would like to know if constructive measure theory has been developed somewhere ? I am more precisely interested in the (constructive) theory of completely continuous valuation on ...
2
votes
0answers
156 views

An inequality for Lp-functions

I am interested in the following inequality: \begin{equation} \int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...
3
votes
0answers
64 views

A small planar set containing a large family of curves

A beautiful construction described in [1] shows a compact connected plane set of measure zero containing circles (circumferences) of every radius between zero and one. A corollary to a theorem proved ...
7
votes
0answers
139 views

Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability ...
1
vote
1answer
83 views

Submodular measures on the hypercube

By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the ...
6
votes
0answers
273 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
2
votes
1answer
153 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
3
votes
1answer
128 views

Weak continuity of Lebesgue decomposition

Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function ...
4
votes
1answer
183 views

Doubt on Morrey spaces of measures according to T. Giga and Y. Miyakawa

I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the ...
2
votes
0answers
85 views

convolution of surface measures

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure ...
1
vote
1answer
221 views

A set of positive-measure not being a countable union of cylinder sets and zero-measure sets?

Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ ...
16
votes
2answers
571 views

A moment problem on $[0,1]$ in which infinitely many moments are equal

Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$. Let their $n$-th moments be denoted by $\mu_n$ and $\nu_n$, respectively, for $n \in \mathbb{N}$. If we know that $\mu_n=\nu_n$ for ...
2
votes
0answers
180 views

Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated. 1) It is the ...
2
votes
1answer
260 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
21
votes
2answers
1k views

Most significant results in motivic integration theory?

I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, ...
0
votes
2answers
195 views

Two questions about Convex Sets and Lebesgue Measure

For any positive integer n, let E(n) denote n-dimensional Euclidean Space and let L(n) denote n-dimensional Lebesgue Measure on E(n). Take n=2 for simplicity. There are uncountably many convex subsets ...
4
votes
1answer
200 views

Coupling of non-probability/sub-probability measures

A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...
10
votes
1answer
456 views

Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normed) probability measure ...
6
votes
2answers
117 views

Vanishing of integral on hemispheres implies vanishing of function?

Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ ...
4
votes
0answers
201 views

decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
1
vote
1answer
100 views

On the existence of a signed measure related to the Hausdorff moment problem

This M.O question reminded me of a problem I've seen a while ago and was not able to solve. Any reference about this kind of problem would be nice. The problem is the following : Does there exist a ...
4
votes
1answer
145 views

On vanishing signed measure

It is quite easy to show that if $\mu$ is positive finite Borel measure on, say $[0,1]$, and for all $n \in \mathbb{N}$ $$\int_{[0,1]} e^{-nx}\mu(dx)=0$$ holds true, then $\mu=0$. Does this still ...
8
votes
0answers
346 views

Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a ...
1
vote
0answers
153 views

When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure. Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
4
votes
1answer
129 views

Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
3
votes
0answers
78 views

Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
3
votes
1answer
174 views

Extending Tarski's Theorem on invariant measures

Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$. I am ...
3
votes
1answer
323 views

A reference for this possibly well-known fact concerning the Kakeya conjecture?

I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of ...
1
vote
2answers
346 views

Reference request: learn measure theory for PDEs

I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...
0
votes
0answers
138 views

Is there a translation invariant measure on an infinite dimensional space 'without points'?

This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...
0
votes
1answer
137 views

identically distributed random variables and measure-preserving transformations

Let $X$ and $Y$ be identically distributed bounded random variables defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$. I want to know if there always exists an invertible ...
4
votes
1answer
246 views

Winning sets of full measure (Schmidt's game)

A quick reminder of the definition of Schmidt's game: Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let ${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball ...
1
vote
1answer
183 views

Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful. Let $V \subset H ...
10
votes
2answers
677 views

Why do we want maps to be measurable (in countably-additive setting)

When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...
5
votes
0answers
205 views

Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete separable metric spaces. Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...
4
votes
1answer
356 views

Are there uniformly discrete paradoxical sets in $R^3$?

I think there aren't any discrete paradoxical sets in $R^2$ (any isometry that mapped a discrete set into itself would have to either be a glide-reflection, a translation or a rotation by $2\pi/n$, ...
6
votes
2answers
242 views

Borel kernel over an analytic set implies existence of a Borel map

Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel ...
3
votes
1answer
125 views

Maps that are a.e. equal have almost the same graphs

Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...
2
votes
1answer
145 views

Universally measurable map coincides a.e. with a Borel map

Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set ...
6
votes
1answer
510 views

Do Measurable Cardinals Exist? (assuming ZFC)

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ...
3
votes
1answer
220 views

Conditional probabilities for all pairs of subsets in $\mathbb R^2$?

Parikh and Parnes showed that one can define a isometry-invariant finitely-additive conditional probability for all pairs of subsets of the interval. Can one extend this result to bounded subsets of ...
4
votes
1answer
220 views

Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
4
votes
3answers
510 views

What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions: 1) Let $X$ and $Y$ be two measure spaces. Suppose for every point $x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full measure in $Y$. Suppose $V \subset ...
0
votes
0answers
229 views

Measure induced by function

It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also ...
5
votes
1answer
474 views

Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one. $I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...
3
votes
1answer
212 views

Prohorov's theorem for random elements of Hilbert space: weak convergence

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...
3
votes
1answer
193 views

Conditional law as a random measure and convergence of random measures

I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...
13
votes
3answers
620 views

How do we express measurable spaces using type theory?

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...
11
votes
3answers
600 views

What fraction of n-point sets in the unit ball have diameter smaller than 1?

This question is inspired by a recent talk by Matt Kahle on random geometric complexes. Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
1
vote
2answers
184 views

An almost orthogonality principle for L^p

I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it goes: If two ...