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

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3
votes
1answer
35 views

Is every regular Borel outer measure topologically additive?

If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive? If so what is a proof or a counterexample? Definitions: Topologically Additive: $X$ is a topological space, $m$ ...
-1
votes
0answers
61 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
0
votes
0answers
23 views

Random Variables with Simply Connected Support [closed]

Let $X$ be a random variable with simply connected support on real line. Define $Z_n=\sum_{i=1}^n X_i$ and $X_i\sim X$ for all $i$. Does $Z_n$ has simply connected support, for all $n\geq 1$?
0
votes
1answer
108 views

Necessary and sufficient conditions for Kolmogorov's Extension Theorem

Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\...
0
votes
1answer
98 views

Doubling metrics, doubling measures, Lebesgue density

As stated in this question, Lebesgue differentiation theorem holds on locally doubling space? and proved here, http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf the Lebesgue differentiation theorem (...
0
votes
1answer
81 views

A uniform Lebesgue density theorem

The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define $$ \chi_{A,r}(x)=\frac{\mu(A\cap B_r(x))...
0
votes
0answers
42 views

Conditions for supremum and conditional Expectation to commute

I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...
0
votes
0answers
68 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
1
vote
2answers
91 views

Existence of a Borel measurable function below any positive function

Let $f\colon \mathbb R\to (0,\infty)$ be a function taking positive values. Does there exist a Borel measurable function $g\colon \mathbb R\to (0,\infty)$ taking positive values as well such that $g(x)...
3
votes
1answer
126 views

Does the Lebesgue measure induce a finitely additive measure on the Boolean algebra of regular open subsets of (0,1)?

Let $(0,1)$ the unit interval. An open subset $\mathcal{R}\subseteq(0,1)$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. Unfortunately, ...
1
vote
1answer
145 views

Finitely additive measures on Boolean algebras of regular open subsets: Is there a relationship with Borel measures? A theory of integration?

Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. ...
4
votes
2answers
239 views

Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$: $$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
1
vote
1answer
76 views

Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
2
votes
1answer
77 views

Meaningful formalization of a continuum of Bernoulli random variables [closed]

I was wondering if there is a meaningful formalization for a continuum of Bernoulli random variables. Informally speaking, consider the interval $[0,1]$, and let's say that for every $x \in [0,1]$, ...
1
vote
0answers
149 views

A certain measure on Banach algebras

According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras". Is there a reference who introduce the following measure on ...
1
vote
0answers
137 views

Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence

Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let $$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$ be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...
6
votes
3answers
177 views

A question on invariant measures

Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures. I do not know is there some nice functional characterization of the following set $\{...
1
vote
1answer
156 views

Could I affirm that $f$ is not identically 0?

Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric: $ d(x,y)=\sum_{i\geq 1}\frac{|...
1
vote
1answer
96 views

Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
-1
votes
0answers
99 views

Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...
6
votes
4answers
504 views

Speed of convergence in Lebesgue's density theorem

Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit $$\lim_{\epsilon\...
12
votes
1answer
334 views

Does there exist a ``continuous measure'' on a metric space?

Let $X$ be a separable complete metrizable space. Does there exist a complete metric $d$ and a Borel measure $\mu$ such that (a) $\mu(B_r(x))<\infty$ for every open ball $B_r(x)$ of radius $r>...
1
vote
0answers
77 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). Let $\...
2
votes
0answers
67 views

Characterizing the optimimum over the space of probability measures

Consider the following optimization problem: \begin{equation} \max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x) \end{equation} where $\mathcal{M}$ is the space ...
1
vote
1answer
98 views

Results for Hausdorff Measure after Linear Transformation

For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with $$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
4
votes
1answer
66 views

Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
3
votes
0answers
44 views

Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions. We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
4
votes
1answer
76 views

sequence of graphs converge in the sense of varifold to multiplicity 2 plane

Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
3
votes
2answers
146 views

How many cuts are required for a weighted-proportional cake-cutting?

In proportional cake-cutting, there are $n$ agents with equal entitlements to a "cake" (an interval). Each agent $i$ has a nonatomic value measure $V_i$ over the cake, and it is required to create a ...
1
vote
0answers
70 views

Tensor product of sigma-algebra?

Does there exist an explaination of the product of 2 sigma-algebra in terms of of tensor product ? Which could explain the tensor product sign for the product of 2 sigma-algebra. In particuliar, does ...
5
votes
0answers
179 views

Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
3
votes
1answer
121 views

$\nu$ is a Dirac delta

Let $X$ be an locally compact Hausdorff space and $m$ a positive regular Borel probability measure where $m(Y)$ is 0 or 1 for any Borel set of $X$. Does it necessarily follow that $m$ is a Dirac delta?...
1
vote
0answers
66 views

Reference for a simple fact in measure theory (semi-algebras)

What is the textbook where the following simple fact can be found: A measure defined on a semi-algebra S can be extended to a sigma-algebra generated by S. In the texbooks that I have looked into ...
1
vote
1answer
76 views

Metrizability of the space of probability measures endowed with the topology of setwise convergence

Let $X$ be a separable completely metrizable space, let $\mathscr{B}(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathscr{P}(X)$ denote the space of all probability measures on $(X, \...
0
votes
0answers
47 views

The density one properties of $\mathcal H^{N-1}$-rectifiable set

Let $S\subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set. Then we know that there exist countably many Lipschitz $N-1$-graphs $\Gamma_i\subset \mathbb R^N$ such that $$ \mathcal H^{N-1}\...
6
votes
2answers
234 views

Can a nowhere differentiable function preserve measurability?

I want to know whether a continuous nowhere differentiable function $f: \mathbf{R} \to \mathbf{R}$ can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know ...
1
vote
0answers
58 views

Converge of measures [closed]

Good night, we have the following: Let $(X,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
7
votes
1answer
187 views

Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...
0
votes
0answers
34 views

Generalizing Integration by parts for general bounded continous measure

Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...
2
votes
1answer
187 views

Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points? Example: Let $f(x)=\...
0
votes
1answer
168 views

Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting: $$ \...
4
votes
1answer
233 views

Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function. I would need to know something similar for an ...
6
votes
1answer
188 views

Definitions of Hilbert Bundles

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
1
vote
1answer
64 views

Set of General Linear Position with Nonzero Measure

I came to the following question, but I don't have quite a good idea how to approach. Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position? I believe that,...
0
votes
0answers
58 views

Lower bound on the diameter of a ball contained in the stable manifold of a critical point

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Consider the negative of the gradient ...
1
vote
0answers
21 views

Bivariate integration with the range of one variable shrinking to a point

Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration $$ \...
4
votes
2answers
222 views

Error estimate in the spectral theorem of compact operators on a Hilbert space

Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
4
votes
1answer
89 views

Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, $D \subseteq X \times A$ be an analytic set, and $D_x := \{a \in A : (x, a) \in D\}$ denote the $x$-section of $D$ at $x \in X$. ...
4
votes
1answer
73 views

Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions

Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
2
votes
1answer
86 views

Measurability of integrals with respect to different measures

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...