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

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6
votes
2answers
195 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
48 views

Converge of measures [on hold]

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
102 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
28 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 ...
5
votes
0answers
153 views

Dual of $BV_0(\Omega)$

It is previously pointed out in Dual or pre-dual of BV that the dual of $BV_c(\Omega)$ (BV functions with essentially compact support in $\Omega$) are so called strong charges, i.e. distributions for ...
2
votes
1answer
173 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 ...
0
votes
1answer
159 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
226 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 ...
-1
votes
0answers
41 views

conformal measures - definitions, examples and references

I would like to know the exact definition of conformal measure in compact and non compact spaces and not compact, as well as examples of conformal measures Also, if someone can refer me to literature ...
6
votes
1answer
176 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
61 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 ...
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
19 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
218 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 ...
3
votes
1answer
63 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
70 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
85 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 ...
2
votes
0answers
254 views

Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
2
votes
2answers
196 views

How to generalize normal number theorem

The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
4
votes
1answer
262 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
0
votes
1answer
122 views

Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying $$\int_{\mathbb ...
1
vote
1answer
120 views

Integration against Borel measures on compact Hausdorff spaces

I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof. Suppose that $X$ ...
3
votes
0answers
140 views

A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: ...
3
votes
1answer
125 views

A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$

I asked this at math.stackexchange, but nobody answered. Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ...
14
votes
1answer
475 views

Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory provides a robust, unifying theory that organizes most of the classification problems of classical mathematics into a hierarchy, allowing us ...
2
votes
1answer
150 views

Non-completeness of the Borel-Lebesgue measure and countable choice

Is it possible to prove the non-completeness of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) without the full axiom of choice, but still with Countable Choice ...
1
vote
0answers
27 views

Pointwise convergence of a sequence of approximate limits of BV functions

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
5
votes
0answers
240 views

Equivalence of the Banach-Tarski paradox

I am working on the Banach-Tarski paradox and the fact that the Hahn-Banach theorem implies that paradox. The proof involves the equivalence of the Hahn-Banach theorem and the fact that for every ...
3
votes
1answer
291 views

Invariant probability on a unit ball of a Banach space

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries. Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
5
votes
1answer
314 views

Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions: Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and ...
0
votes
0answers
69 views

Hausdorff dimension and Hausdorff measure

Let us consider a curve $E\subset \mathbb{R}^2$ which is a $(\delta,R)$- Reifenberg flat domain and suppose also that it holds the following estimate on the Hausdorff dimension:$\mbox{ ...
1
vote
2answers
172 views

Is there a name for this metric on a Borel sets

Consider a finite measure space $(X,\Sigma,\mu)$. Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by $$d(\sigma_1,\sigma_2) = \mu \left\{ (\sigma_1^c \cap \sigma_2) \cup (\sigma_1 \cap ...
3
votes
1answer
93 views

Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$ Here ...
0
votes
1answer
126 views

Countably generated $\sigma$-algebra

Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space? I assume not, so here is a more ...
3
votes
2answers
117 views

Extreme couplings

Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...
0
votes
0answers
71 views

Measurable sets of probability measures $\{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}$

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the ...
6
votes
1answer
224 views

Completion of spaces of measures w.r.t. weak norms

For the sake of concreteness denote by $M_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x_0\in X$. On this space ...
12
votes
2answers
259 views

Are finitely generated amenable groups positively finitely generated?

Let $G$ be a finitely generated amenable group. Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability? Being more formal, note that $G^n$ is ...
0
votes
0answers
88 views

Bounds on Wasserstein (Kantorovich) distance

Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...
2
votes
1answer
119 views

Visualizing ANOVA Decomposition [closed]

Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...
3
votes
2answers
133 views

Convex combinations of Bernoulli Measures

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures? I mean, we are in the symbolic space ...
3
votes
0answers
44 views

measure of an image under an argmax function

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function. For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
1
vote
0answers
113 views

Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan. In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
0
votes
1answer
66 views

Approximating characteristic functions by cutting the real axis into smaller and smaller pieces

Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
3
votes
0answers
96 views

Radon-Nikodym derivative as a limit of ratios

This question is related to Radon-Nikodym derivatives as limits of ratios. Let $F$, $G$ be sigma-finite measures (or at least probability measures) on $\mathbb{R}$ such that $F \ll G$. The theorem ...
1
vote
0answers
63 views

Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$? For examples: are there: Ito ...
2
votes
0answers
47 views

Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...
0
votes
0answers
66 views

Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
4
votes
3answers
192 views

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$. ...
2
votes
1answer
75 views

Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem: \begin{equation} \mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...