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

learn more… | top users | synonyms

2
votes
0answers
25 views

Sequences of transition probability measures

Suppose that $X$ and $Y$ are compact metric spaces. A Borel probability measure $\mu$ on $X\times Y$ satisfies $$ \mu(A\times B)=\int_A\mu(B|x)\mu_X(dx), $$ for $A$ and $B$ Borel sets in $X$ and $Y$ ...
4
votes
2answers
586 views

Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish? I have a specific space in mind, so if the ...
5
votes
1answer
59 views

Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
5
votes
2answers
70 views

Accuracy of the truncated Hausdorff moment problem

For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that $$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$ In other words, $M_s$ ...
0
votes
1answer
49 views

Convergence of measures to an absolutely continuous measure

Suppose that $\{\mu_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ ...
7
votes
3answers
237 views

Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...
0
votes
1answer
83 views

topology of setwise convergence of measures

It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a ...
3
votes
1answer
85 views

Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...
8
votes
1answer
122 views

Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$? For ...
-2
votes
0answers
37 views

$L_{\infty}$-norm of a $\delta(t)$-“function”? [closed]

In different contexts the $L_\infty$- norm may sometimes be defined as the essential least upper bound $\|\cdot\|_\infty=\operatorname{ess} \sup_t |\cdot |$ and sometimes as just the least upper bound ...
0
votes
2answers
100 views

When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix $$ N(y,f) = \#\{x\in X \mid f(x) = y\} $$ is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then ...
0
votes
0answers
112 views

an existence question from topological groups [migrated]

$G$ is a topological group, $A$ and $B$ are the subsets of $G$, we denote $AB$=$\{ab:a \in A, b\in B \}$. Let $G$ be a locally compact Hausdorff topological group, $m$ is a left Haar measure on $G$, ...
0
votes
0answers
14 views

Convergence in $L^1$ for characteristic functions [migrated]

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...
6
votes
1answer
273 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
7
votes
1answer
266 views

Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition. (EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...
3
votes
2answers
191 views

When does a function space allow for point evaluations? [closed]

Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology. What are necessary and sufficient conditions for point evaluations at arbitrary $x ...
1
vote
0answers
47 views

Pollard's construction of measures from set functions on lattices of sets

Theorem 12 in Appendix A of Pollard's A User's Guide to Measure Theoretic Probability gives conditions under which a set function defined on a family of sets $\mathscr{K}$ which is closed under finite ...
-1
votes
1answer
99 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
0
votes
0answers
39 views

Uncountability of the set of subsets of $\mathbb N$ [migrated]

Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for every pair of elements in $E$, ...
7
votes
0answers
184 views

A question about finitely additive extensions of Lebesgue measure

Suppose $m:P([0, 1]) \to [0, 1]$ is a finitely additive measure extending the Lebesgue measure. Must there exist some $X \subseteq [0, 1]$ such that $m(X \cap I) = |I|/2$ for every sub interval $I ...
2
votes
0answers
70 views

convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that $$\int f d \mu_n \to \int f d\mu$$ ...
2
votes
1answer
63 views

Quotient sigma-algebra generated by quotient-measurable generating sets

Let $X$ be a measurable space whose $\sigma$-algebra is generated by a family $\mathcal{G}=\bigcup_n \mathcal{G}_n$ of subsets of $X$, where $(\mathcal{G}_n)$ is a sequence of $\sigma$-algebras on $X$ ...
10
votes
1answer
210 views

Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...
3
votes
2answers
158 views

Plane measurable sets and measurable rectangle

Does every measurable set in the plane with positive Lebesgue measure contain a cartesian product of two measurable sets of the real line with positive Lebesgue measures?
4
votes
1answer
174 views

Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...
0
votes
0answers
78 views

Alternate proof for Caratheodory extension theorem

This question is on the intuition behind the Caratheodory definition of measurable sets as given in Billingsley. He motivates by saying that we "should" call a set $A$ measurable if $$P^*(A) + ...
2
votes
1answer
138 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
3
votes
2answers
236 views

Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$ This question is strongly linked to is the space of all borel measures on ...
0
votes
1answer
92 views

Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that ...
11
votes
2answers
272 views

A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
6
votes
2answers
203 views

Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and g is radially symmetric, the function $ (0, \infty )\ni t \mapsto g ...
1
vote
0answers
75 views

Outer measure preserving bijection

Suppose X is a Sierpinski set (So X is uncountable and every null subset of X is countable). Let f be a bijection on X. Must/Does there exist a non null subset Y of X such that for every subset W of ...
2
votes
1answer
223 views

Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
2
votes
1answer
97 views

Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...
23
votes
2answers
426 views

Mid point free sets

Given a subset X of unit interval, can we find a subset Y of X of same outer measure as X such that Y does not contain three points of the form x, y and (x+y)/2? I can do this assuming CH but can we ...
4
votes
3answers
374 views

Is the space of all borel measures on $\mathbb R^n$ isomorphic to the tensor product of spaces of borel measures on $\mathbb R$?

Similarly to the decomposition $L_2(\mathbb R^n) = L_2(\mathbb{R})^{\otimes n}$ as vector spaces (and even as Hilbert spaces) , do we have $bm(\mathbb{R^n}) = bm(\mathbb{R})^{\otimes n}$ where ...
2
votes
2answers
114 views

On the mesurability of a VItali set w.r.t. a Lebesgue absolutely continuous measure

I've seen two kinds of demonstrations of Vitali's sets being not measurable (for example, answer number 2 here: http://math.stackexchange.com/questions/137949/the-construction-of-a-vitali-set, I ...
4
votes
3answers
318 views

Measure with `somewhere dense' support

Let $X$ be a compact Hausdorff (but not necessarily metrizable) space. Is it always true that there exists a probability Borel measure $\mu$ and an open set $U$ such that any nonempty open set ...
4
votes
1answer
91 views

Existence of doubling non-Polish metric measure spaces

Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < ...
3
votes
2answers
174 views

Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions? [closed]

Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int_{\mathbb R}f\,d\mu=\int_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions ...
3
votes
0answers
129 views

Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
2
votes
1answer
114 views

Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...
3
votes
1answer
81 views

Criteria for Compactness of a Closed in $L^2$ Spaces [closed]

$(X, \mathcal{B}, \mu)$ is a measure space. Is there any well-known criteria for compactness of a closed set in $L^2(X, \mu)$? If the answer is negative what about $L^2(\mathbb{R}^n,\mu)$(in this ...
4
votes
1answer
179 views

Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure? Of course if such ...
4
votes
2answers
166 views

How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
0
votes
0answers
74 views

absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
2
votes
1answer
171 views

Reading Ratner's paper “Ragunathan's conjectures for SL(2,R)”

Hello everyone (interested), I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of ...
3
votes
1answer
338 views

Notation: Categories of measur(abl)e spaces

Is there a common notation in the literature for the category of measurable spaces and measurable maps? the category of measure spaces and measure-preserving maps? The nlab suggests ...
3
votes
2answers
203 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
0
votes
1answer
54 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...