# Questions tagged [measure-theory]

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

**5**

votes

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176 views

### Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...

**1**

vote

**0**answers

60 views

### Nonlinear maps in Riesz Thorin theorem

The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...

**3**

votes

**0**answers

58 views

### Dependency of the Wasserstein distance on the parameter: a differential perspective

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...

**0**

votes

**0**answers

71 views

### On measurability in Wiener space

Let $f$ be a complex-valued continuous function on Wiener space such that $|f|$ is measurable. Is $f$ then measurable, too? I am looking for a proof or a counterexample.

**7**

votes

**2**answers

241 views

### Non-separable metric probability space

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...

**3**

votes

**0**answers

38 views

### Measure of set of vectors whose outer product are bounded

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.
Define the map $v_k: E^k \rightarrow \mathbb{R}$ that
sends a $k$-uple $x_1,\cdots, x_k$ of ...

**1**

vote

**1**answer

91 views

### are there measure preserving mapping in this case?

Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...

**3**

votes

**1**answer

249 views

### Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...

**0**

votes

**1**answer

104 views

### Questions on a new definition of continuous multivariate distribution

For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am ...

**1**

vote

**2**answers

178 views

### Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?

Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...

**9**

votes

**1**answer

255 views

### Can a big set always look small?

For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle.
Say ...

**2**

votes

**1**answer

86 views

### Closedness of the set of probability measures anihilating a measurable function

Let $X$ be a compact metrizable topological space and let $f$
be a bounded, real valued, Borel function on $X$. Denoting by
$P(X)$ the collection of all probability measures on $X$,
consider the ...

**1**

vote

**1**answer

90 views

### Convergence of measurable functions in a locally compact space

Set $(X,\mathcal{B})$ a measurable space. If $f:X\rightarrow[0,\infty)$ is a measurable function then exists a sequence of simple functions $\{s_n\}_{n\geq1}$ such that
$$0\leq s_1 \leq s_2\leq \...

**2**

votes

**1**answer

115 views

### Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...

**4**

votes

**0**answers

96 views

### A quantity that distinguishes finer than Hausdorff dimension

Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...

**2**

votes

**0**answers

102 views

### Probability bound involving random, convex sets

Let $X,Y$ and $Z$ be random vectors in $\mathbb{R}^d$ defined on $(\Omega,\mathcal{H},\mathsf{P})$ s.t. $Z$ is $\mathcal{F}$-measurable for some $\mathcal{F}\subset\mathcal{H}$. Define a family of ...

**2**

votes

**1**answer

110 views

### Support of functions in Fourier domain

Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...

**4**

votes

**2**answers

167 views

### Can the differential entropy of a continuous distribution with lebesgue integrable density be negative infinity?

Define the (differential) entropy for a density $f$ as
$$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$
I am trying to find a $f \in L_{1}([0,1])$ such that $f\geq 0, \int_{0}^{1} f(x) dx = 1$...

**7**

votes

**0**answers

84 views

### When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?

Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...

**5**

votes

**2**answers

167 views

### Von Neumann's theorem on realizing automorphisms of the measure algebra

I'm looking for a proof, in English, of the following theorem due to von Neumann (which apparently originates in the paper Einige Sätze über Messbare Abbildungen, Ann. of Math, 1932):
Every ...

**1**

vote

**1**answer

91 views

### Borel $\sigma$-algebra on the space of Hölder continuous functions

Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...

**3**

votes

**1**answer

75 views

### Antisymmetry of the stochastic order

An ordered topological space is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ ...

**2**

votes

**2**answers

235 views

### Non-probabilist term for conditional expectation?

When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about ...

**2**

votes

**0**answers

102 views

### Parametric distances on product spaces of measures

Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...

**1**

vote

**0**answers

50 views

### Are the sets whose convex hull surface admits multiple representations a shy set of sets?

Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...

**9**

votes

**0**answers

213 views

### Covering inequality for sets of intervals

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point ...

**4**

votes

**1**answer

83 views

### automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...

**1**

vote

**1**answer

125 views

### How small can a set admitting a nonatomic finite measure be?

Is it consistent that there exists a nonzero atomless finite measure
on some $\sigma$-algebra on a cardinal $\kappa$ satisfying $\kappa<\mathfrak{c}$? Can
there be such a measure on $\omega_1$ ...

**4**

votes

**0**answers

47 views

### $L^p$-spaces for locally convex spaces

Let $(X,\sigma)$ be a locally convex space, say generated by a family of seminorms $\mathfrak{P}$. I know that there is the notion of the space of integrable functions $f:\Omega\rightarrow(X,\sigma)$ ...

**1**

vote

**1**answer

46 views

### Infinitely many independent functions that are only frequency localized?

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...

**4**

votes

**1**answer

99 views

### Non-linear translation invariant functionals on $L^1$

I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...

**8**

votes

**1**answer

199 views

### Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.

**2**

votes

**1**answer

356 views

### How “compact” are sets of finite measure?

Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover.
Now consider the following situation:
Everything I say in the following is with respect to the ...

**1**

vote

**0**answers

956 views

### The Rise and Fall of Dictators & How it Depends on Our Choice

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively.
Shelah, Saharon, On the Arrow property. Adv. in Appl. ...

**3**

votes

**2**answers

155 views

### Existence of a separating affine functional

Let be $S$ a separable(non compact) metric space and $X=C_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive ...

**3**

votes

**0**answers

121 views

### Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures

Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...

**2**

votes

**2**answers

134 views

### References for the Spectral Theorem ( Multiplication Operator Form)

Let $A_1$ and $A_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\...

**0**

votes

**1**answer

114 views

### Does sequence almost sure convergence imply almost sure convergence?

This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\...

**4**

votes

**0**answers

95 views

### Convergence (topology) for $\sigma$-finite measures

I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by ...

**4**

votes

**0**answers

86 views

### point-wise approximation of the identity in hereditary Lindelof spaces

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...

**1**

vote

**2**answers

89 views

### decomposing a measure into relative atom and atomless parts

Let $(X,\mathcal{X},\mu)$ be a standard probability space. A measure $\mu$ is atomic if $\mu$ is supported on at most countable many atoms, and is atomless if $\mu$ has no atom (a point x is an atom ...

**12**

votes

**3**answers

359 views

### A characterization of $L_1(\mu)$ in $L_\infty(\mu)^*$

Let $\mu$ be a finite positive measure on a set $M$:
$$
\mu(M)<\infty.
$$
As is known, the Banach dual space $L_\infty(\mu)^*$ to the space $L_\infty(\mu)$ contains $L_1(\mu)$, but (excluding some ...

**0**

votes

**0**answers

34 views

### Is volume of abstract polytope realisation bounded by length of edges?

Suppose we have abstract polytope F of dimension d ( that is the greatest rank facet has rank n). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$, and ...

**1**

vote

**0**answers

37 views

### metric density of a set in the plane with respect to distinct metrics

Let $A \subseteq \mathbb{R}^2$ be Borel (or even open or closed), let $\mathbf{0} = ( 0 , 0)$, and let $\lambda$ be the Lebesgue measure in $\mathbb{R}^2$. Let $
\mathcal{D}^2_A ( \mathbf{0} ) = \lim_{...

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vote

**0**answers

67 views

### density of fractal measures

Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...

**4**

votes

**1**answer

111 views

### Weak convergence of measures on dense sets

We are given a complete (separable) metric space $X$ and a dense subset $D\subset X$. Consider a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $$\int\limits_D f_n \, {\rm d}\mu\...

**2**

votes

**0**answers

106 views

### description of dual space of space of Radon measure equipped with topology of weak convergence

Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence":
$$
\mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...

**3**

votes

**0**answers

100 views

### Other than Brownian motion, when else is it possible to define “normalized weighted infinite dimensional Lebesgue measure”?

In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...

**3**

votes

**1**answer

128 views

### What is the Wiener measure of the curves with Hölder index $\frac 1 2$?

One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...

**4**

votes

**1**answer

213 views

### Approximation of the identity by finite range functions in topological vector spaces

Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists ...