Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,897
questions
0
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0
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22
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Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
3
votes
1
answer
96
views
Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
7
votes
2
answers
277
views
Integral means vs infinite convex combinations
Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function.
Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
-4
votes
0
answers
81
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What is the Measure of the Permutations of the Real Numbers? [closed]
Since the permutations of the real numbers would form a set of cardinality $\aleph_2$, do we just say it has infinite measure? It would seem to be the case for a Lebesgue measure.
Is there a standard ...
0
votes
0
answers
20
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
2
votes
2
answers
203
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How should the "measure theoretic" Jacobians of a dynamical map be understood in Lai-Sang Young's "Recurrence Times and Rates of Mixing"
In Young's article: Recurrence Times and Rates of Mixing, she uses multiple times the notation $JF, JF^k, JF^R$ to mean the Jacobian of a dynamical map $F:\Delta\to\Delta$ w.r.t. a given reference ...
0
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0
answers
17
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reference request: product measures defined by a subsequence of measures
Suppose $\{\mu_n\}_{n\in\mathbb{N}}$ is a sequence of pairwise equivalent probability measures, each of which is defined on $\mathbb{R}$. Let $\bigotimes_n\mu_n$ be the product measure defined on $\...
2
votes
1
answer
58
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From convergence of sequences to uniform convergence in probability
For $n=1, 2,\ldots$ consider a sequence of sets of ascending integers $I_n=\{\underline{i}_n,\underline{i}_n+1, \ldots, \overline{i}_n\}$, with $\underline{i}_n \to \infty$ and $\underline{i}_n=o(\...
2
votes
0
answers
39
views
If a probability measure is a mixture of products of its marginals, does it have finite moments?
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$. For a linear subspace $E\subset \mathbb{R}^n$, let $\mu_E$ denote the marginal of $\mu$ on $E$. The usual orthogonal complement of $E$ is ...
1
vote
0
answers
30
views
Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
2
votes
1
answer
101
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On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
2
votes
1
answer
89
views
For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
1
vote
0
answers
105
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Density of absolutely continuous measures on a Polish Space
Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
0
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0
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42
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Does local convergence in $L^p$ follow from weak convergence of measures and uniform $L^p$ boundedness?
On page 610 in Sur les mesures de Wigner by P.-L. Lions and T. Paul, the authors claim the following:
Let $\mathcal{M}_{w-*}$ be the space of bounded positive measures on $R^d$, equipped with the weak*...
1
vote
3
answers
160
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Graph on $\mathbb{N}$ where almost every vertex is shy
The following question is loosely based on the friendship paradox.
Let $G=(V,E)$ be a simple, undirected graph. For $v\in V$, we let the neighborhood of $v$ be $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ ...
1
vote
1
answer
104
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A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
1
vote
1
answer
76
views
On the definition of symmetric rearrangement
For a measurable function $u:\mathbb{R}^{n}\to \mathbb{C}$ one usually defines the symmetric rearrangement $u^{*}:\mathbb{R}^{n}\to \mathbb{R}^{+}$ as follows:
\begin{equation*}
u^{*}(x)=\int_{0}^{\...
0
votes
1
answer
67
views
Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$
We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a ...
6
votes
1
answer
209
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Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$
Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
0
votes
1
answer
178
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Does weak convergence in $L^2$ imply convergence a.e. of a subsequence? [closed]
The title pretty much explains it all. Let $u_n\in L^2(\mathbb{R}^n)$ be a sequence converging weakly in $L^2$ to some $u\in L^2(\mathbb{R}^n)$, that is $\int u_k v \to \int u v$ for all $v\in L^2(\...
1
vote
1
answer
93
views
Maximal element w.r.t. abolute continuity of measures
Suppose that $\mu$ is a $\sigma$-finite measure on $\mathcal{X}\equiv\bigotimes_{i=1}^n\mathcal{X}_i$. Let $\Pi$ denote the set of all $\sigma$-finite product measures on $\mathcal{X}$. Define
$$
\...
0
votes
0
answers
90
views
Some stability and estimate of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
0
votes
0
answers
44
views
Existence of derivative of distribution of exponential family?
Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
0
votes
0
answers
22
views
An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
0
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0
answers
171
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The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
4
votes
1
answer
232
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Does a generic linear map admit a vector whose iterates span $V$?
We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$.
Question: ...
0
votes
1
answer
85
views
Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
1
vote
1
answer
91
views
Interchange the deterministic and stochastic integrals
We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
0
votes
0
answers
55
views
Existence of a measurable maximizer
Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
2
votes
0
answers
76
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
2
votes
1
answer
112
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On a density property of signed singular measures
Suppose that $\mu$ is a signed finite Borel measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that
\begin{equation*...
1
vote
1
answer
295
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Is the Borel-Cantelli Lemma applicable here? [duplicate]
Consider $(X_{n})_{n\in\mathbb{N}}$ a sequence of random variables taking values in the set $\mathbb{Z}_{\geq 0}$ where $\mathbb{P}(X_{n} = i) > 0 $ for every $i\in\mathbb{Z}_{\geq0}$ which are ...
8
votes
1
answer
1k
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Worst of both worlds?
It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...
0
votes
1
answer
72
views
Same occupation measure $\Rightarrow$ same trajectory
Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
The occupation ...
18
votes
1
answer
348
views
Is defining measures as functionals ever insufficiently general in practice?
Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here.
The way I learned measures was as set functions on a $\sigma$-algebra with certain ...
0
votes
1
answer
112
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Some continuity issues of the optimal transport map (Brenier map)
Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
1
vote
0
answers
79
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
2
votes
0
answers
139
views
The Hausdorff measure of intersection of annulus and conformal curve
Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...
1
vote
1
answer
130
views
Strongly regular binary sequences
Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A
\subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...
2
votes
2
answers
306
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
9
votes
1
answer
328
views
Disintegration measures and differential forms
Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...
2
votes
1
answer
86
views
Does left-invariance imply right-invariance for closed unimodular normal subgroups?
Suppose we have a locally compact group $G$ and a closed unimodular normal subgroup $N$.
Let $\mu$ be a Borel probability measure on $G$ which is left $N$-invariant.
Does it follow that $\mu$ is right ...
2
votes
2
answers
226
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
0
votes
1
answer
125
views
Relation between the convergence of measures and the evolution of their supports
Let a sequence of Borel probability measures $\mu_n$ converges to $\mu$ in the weak*-topology. Are there any related results on the evolution of the support of $\mu_n$?
For example, in which case, the ...
0
votes
0
answers
65
views
Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?
Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...
0
votes
0
answers
37
views
Weak convergence of Gibbs measures with converging energy functions
Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$
and let $Q$ be a fixed probability measure on $A$.
For each $\theta>0$, define the probability ...
1
vote
1
answer
94
views
Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity
Consider the IVP
$$
\left\{
\begin{aligned}
\frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\
\Phi_n(0,x) &= x && \forall x \in \mathbf{R}
\end{...
1
vote
2
answers
252
views
An integral inequality?
Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...
1
vote
1
answer
113
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
11
votes
3
answers
1k
views
Every function on reals a sum of two surjective real functions?
From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...