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

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3
votes
0answers
105 views

A conjecture about the measure estimates of a trigonometric polynomial [on hold]

Formulation of the Conjecture Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...
0
votes
0answers
71 views

Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
4
votes
1answer
169 views

Proof of the Dunford-Pettis theorem

I would like to know where to find a complete proof of the Dunford-Pettis theorem: A sequence $(f_n)_{n\geq 0} \subset L^1$ is uniformly integrable if and only if it is relatively compact for the weak ...
19
votes
2answers
664 views

construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer. The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...
2
votes
1answer
68 views

Why do rotationally ordered configurations have well defined distributon function?

Let $u=(u_{j})_{j \in \mathbb{Z}}$ where $u_{j}\in \mathbb{R}$ for all $j \in \mathbb{Z}$ be a rotationally ordered configuration i.e. $S_{n,m}u>u$ or $S_{n,m}u<u$ or $S_{n,m}u=u$ where ...
0
votes
0answers
79 views

Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$

My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
3
votes
3answers
199 views

Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?

Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel ...
1
vote
1answer
141 views

Bingham's paper “Finite additivity vs countable additivity” [closed]

I am reading N. H. Bingham's (2010) paper: ''Finite additivity vs. countable additivity''. Page 8, he states: "One area where the distinction between finite and countable additivity shows up most ...
0
votes
0answers
130 views

A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...
4
votes
1answer
105 views

On a.e. approximate differentiability of certain continuous real functions

I have the following question: If $f:[0,1]\to \mathbb{R}$ is a bounded continuous function of $\sigma$-finite variation in sense 1, then is it true that $f$ is approximately differentiable a.e. on ...
5
votes
2answers
264 views

non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...
0
votes
1answer
77 views

Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...
3
votes
0answers
71 views

Non Borel Spaces: Gauge Integral

Question Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology? I'm wondering since it seems as the gauge crucially uses ...
2
votes
0answers
75 views

Borel class of a set of measures

Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding ...
2
votes
2answers
138 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
3
votes
2answers
85 views

Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here. Point ...
2
votes
0answers
59 views

Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
2
votes
0answers
48 views

Stronger version of linearity for functions of measures

Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate ...
3
votes
0answers
85 views

Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
2
votes
1answer
180 views

Inverse of a Borel surjection

Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$. ...
3
votes
1answer
192 views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...
2
votes
0answers
64 views

Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)

I am trying to formulate the measure of event $E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$, where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...
4
votes
1answer
105 views

Analytic enlargement of an analytic set

Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus ...
3
votes
0answers
120 views

Embedding probability spaces in the completion of $[0,1]^K$

Question: Can every probability space $(X,\scr F,\mu)$ be $\sigma$-embedded in the completion of the space $[0,1]^K$ (equipped with a product of Lebesgue measure) for some set $K$? Here, $f:\scr F\to ...
4
votes
0answers
203 views

Why does it seem that $rca=rba$? [closed]

The following paradox has got me stumped. I'm hoping someone can point out the error. Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous ...
3
votes
0answers
64 views

Almost everywhere in a rectangle [duplicate]

I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't. Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
1
vote
2answers
142 views

Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$. My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...
1
vote
2answers
96 views

Volume of normal cone of a simplex (at a vertex)

This question is related to an orthant-type simplex in $\mathbb{R}^n$, which can be defined as $$ S = \{x\in\mathbb{R}_{+}^n: \sum_{i=1}^nx_i \leq 1\}=\overline{\mbox{conv}}(0,e_1,\ldots,e_n). $$ For ...
1
vote
0answers
96 views

Make this marginalization statement rigorous

Intuition tells me that $$ p(x\,|\,y) = \int p(x,\theta\,|\,y) \; d\theta$$ by the "law of marginalization", pretty much for any object $\theta$. I would like to make this statement rigorous, ...
3
votes
1answer
148 views

Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times ...
3
votes
1answer
150 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. ...
5
votes
2answers
180 views

Relationship of Euler product to coprimality densities for arbitrary sets of primes

Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...
1
vote
1answer
45 views

Additivity of asymptotic density of periodic sets

It is well known that the asymptotic density of an infinite union of disjoint sets of integers may not be the sum of their individual asymptotic densities. Can this failure of countable additivity ...
3
votes
1answer
112 views

For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1

This is related to a question asked on mathstackexchange http://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
1
vote
2answers
215 views

Question on separability of a measure

Following this question here this question come to mind. Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of ...
1
vote
1answer
48 views

Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
0
votes
0answers
45 views

Approximate of binary function by indicator functions of one variable

Let $(X,F,u)$ and $(Y,G,v)$ be two probability spaces. I know a result that the indicator functions of product measurable sets can be approximated by indicator functions of one variable. That is, for ...
2
votes
0answers
50 views

Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as ...
6
votes
2answers
98 views

How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure?

It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to ...
1
vote
1answer
115 views

Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm. The Laplace-Belrami-Operator $\Delta$ on $X$ with ...
6
votes
1answer
135 views

Minimal generator of an algebra or a sigma-algebra

I may be asking a trivial question, but I am a bit confused about it. I have tried to search for the concept of a minimal generator of an algebra or a sigma-algebra on a set, but have found this ...
2
votes
0answers
33 views

Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction. Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
0
votes
1answer
171 views

A very natural question in weak* topology [closed]

Can you provide me a counter example for this. Suppose that I have a sequence of probability measures $(\mu_{r,t})_{r,t>0}$ on a compact space metric $X.$ Suppose additionally that: there exists ...
3
votes
1answer
83 views

Metric density theorem in most general setting?

It's a consequence of Lebesgue's theorem that every measurable $E\subset\mathbb{R}^n$ has a metric density that's $1$ a.e. on $E$ and $0$ a.e. on $\mathbb{R}^n\setminus E$. What are the most general ...
1
vote
1answer
111 views

Ergodic decomposition and integral representation of functions that depends on a measure

Let $X$ be a compact metric space, $T:X \to X$ continuous, $M_T(X)$ the set of borel measure that are $T$-invariant and $E_T(X)\subseteq M_T(X)$ the set of ergodic measures. The ergodic decomposition ...
4
votes
2answers
265 views

The borel $\sigma-$algebra of the set of probability measures

Let $X$ be a compact metric space and $M(X)$ the set of all Borel probability measures on $X$. It is know that $M(X)$ is a convex compact metric space endowed with the weak-* topology i.e. $(\mu_n)_n ...
8
votes
1answer
173 views

Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...
1
vote
0answers
79 views

On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...
3
votes
1answer
170 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
5
votes
0answers
167 views

Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...