**6**

votes

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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](:= ...