Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
803
questions with no upvoted or accepted answers
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Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,588
4
votes
0
answers
173
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Continuous doubling weight vanishing on set of positive measure?
If $I$ is a bounded interval in $\mathbb{R}$, let $2I$ denote an interval with the same center point but double the length.
A doubling measure on $\mathbb{R}$ is a (non-trivial, locally finite, Borel)...
- 41
4
votes
0
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95
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Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
- 1,837
4
votes
0
answers
386
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Topology on the space of Borel measures
Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
- 541
4
votes
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185
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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 $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 3,992
4
votes
0
answers
735
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Radon-Nikodym derivative as a limit of ratios
This question is related to Radon-Nikodym derivatives as limits of ratios.
Let $F$, $G$ be sigma-finite measures (or at least probability measures) on $\mathbb{R}$ such that $F \ll G$.
The theorem ...
- 41
4
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answers
1k
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Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
- 41
4
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307
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Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,267
4
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171
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Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets
Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
- 429
4
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90
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Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
- 1,605
4
votes
0
answers
177
views
Beck-Chevalley for measures?
A measurable set is a pair $(X,\Sigma)$ where $X$ is a set and $\Sigma$ is a $\sigma$-algebra on $X$. The elements $U\in\Sigma$ will be considered as subsets $U\subseteq X$. A morphism of measurable ...
- 8,559
4
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233
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Convergence of probability measures on a generating field of a sigma-field
Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, ...
- 51
4
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222
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A question about measures on groups
Let $G$ be a finitely generated (in my case also amenable) group and $f:G\to[0,1]$. Suppose that there is a finitely additive probability measure $\mu$ on $G\times G$ and a real number $L$ such that $\...
- 5,904
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476
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Two standard probability spaces
Let $(X,\cal{A},\mu)$ be a standard (Lebesgue-Rokhlin) space with complete probabilistic measure (for example, $[0,1]$). Let $\cal{B}\supset \cal{A}$ be a wider then $\cal{A}$ $\sigma$-algebra on $X$, ...
- 103k
4
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211
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The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
- 8,559
4
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440
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Measure Theoretic view of Hardy Littlewood Circle Method
Is it possible to view the Hardy-Littlewood Circle method as the Fourier transform with respect to the Lebesgue measure on [0,1) for an appropriate generating function defined in terms of additive ...
- 41
4
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287
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Weak*-continuity of regular conditional probabilities "in time"
Let $(\Omega, F, (F_t)_{t\geq 0}, \mathbb{P})$ assume that $(X_t)_{t\leq T} $ is some cadlag, real valued stochastic process, not too bad: say something like a Brownian Motion and some Poisson finite ...
- 278
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1k
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Sigma-algebras on Banach Spaces.
I am very interested in counterexamples when cylindrical sigma-algebra is not equal to the borel. I read
link text and
link text.
Especially interessted in l-infinity space. I've read Talgat and ...
- 179
4
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848
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For what sub-$\sigma$-algebra are these two measures equivalent?
In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
- 2,721
3
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179
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Reverse-mathematical strength of Banach-Tarski
What is the reverse mathematical strength of the Banach-Tarski paradox?
The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...
- 1,298
3
votes
1
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237
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Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that
\begin{equation}
\int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty
\end{equation}
...
- 2,749
3
votes
0
answers
150
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Local dimension of stationary measures for iterated function systems with an expanding map
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P = (p/2,p/2,1-p),$ where:
$f_1,f_2: I\to I$, where $...
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answers
137
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Which is the smallest $\sigma$-algebra that contains all analytic sets?
Let $X$ be a polish space. Is the smallest $\sigma$-Algebra containing all analytic sets of $X$ (i.e. all subsets $A \subset X$ which are the continuous image of a polish space) the $\sigma$-algebra ...
- 233
3
votes
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138
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
3
votes
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87
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Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
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156
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Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?
I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
- 204
3
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137
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Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?
It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
- 460
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106
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Is (the generalised) Sard's theorem optimal?
As mentioned in this question (https://math.stackexchange.com/questions/416607/show-that-fc-has-hausdorff-dimension-at-most-zero/446049#446049), in 1965 Sard proved the following result (paraphrased ...
- 460
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140
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Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
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106
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Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
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Is it possible to reconstruct the universally measurable sets in X from the $C^*$-algebra $C(X)^{**}$?
This continues my question of two months ago. Let $X$ be a compact Hausdorff topological space. We consider the $C^*$-algebra $C(X)$ of continuous functions on $X$, its dual space $C(X)^{*}=M(X)$ of ...
- 7,274
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244
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Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 65
3
votes
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answers
219
views
Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,274
3
votes
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209
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Characterization of a Bochner/strongly measurable function solely as a random element
Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively.
This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
- 31
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votes
0
answers
206
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The baker problem
Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
- 4,842
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votes
1
answer
390
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Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
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Is the singular value decomposition a measurable function?
$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators
$$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$
where $\mathbb U_n$ is the ...
- 201
3
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152
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Extension of normal vector field to a domain
Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...
- 31
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answers
124
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Isoperimetric inequality for general metric space
Consider some space $\mathcal{S}$ with metric $d$ and measure $\mu$.
For arbitrary set $H$ denote the $v$-bound of $H$ by $\delta_v(H):= \{x \mid x \notin H: \exists y \in H \text{ s.t. } d(x,y) \le v ...
- 2,683
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votes
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253
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Universally measurable but not analytically measurable set
is there a concrete example (or even a non-concrete example) of a universally measurable set which is not analytically measurable (i.e., not in the sigma-algebra generated by the analytic sets)?
- 119
3
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0
answers
131
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The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)
Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside
step function which has a jump 1 at $t=0$ (it ...
- 369
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votes
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answers
245
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Density of signed measures in dual space
Let $B$ be a Banach space of functions on a Radon space $X$. By the Hahn-Banach theorem, we know that the canonical evaluation map is isometric. That is, for every $f \in B$, we have
$$\|f\| = \sup_{\...
- 103
3
votes
0
answers
168
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Volume of a set vs volume of its projections
Let $V\subset\mathbb R^n$ be "nice" (measurable or Borel or open or convex...) and let $V_{\{i,j,...\}}$ be the projection of $V$ on the subspace spanned by $e_i,e_j,...$.
It is easy to see ...
- 165
3
votes
0
answers
185
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Which stationary varifolds have non-integer density?
A central object in geometric measure theory are the generalised, and weakly defined minimal surfaces called stationary varifolds. Let me recall some definitions. Given an open subset $U \subset \...
- 4,968
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votes
0
answers
665
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Simple proof of the Lebesgue density theorem in $\Bbb{R}^n$
[I posted this on MSE a while ago, but no answer was forthcoming.]
I am looking for a simple proof of the Lebesgue density theorem for $\Bbb{R}^n$. The Wikipedia page on the Lebesgue differentation ...
- 859
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votes
0
answers
102
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Regular Lagrangian flows on a domain of $\mathbb{R}^d$ with a boundary
I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space).
Here, ...
- 161
3
votes
0
answers
96
views
Pettis vs. Dunford integrability of operator valued functions
Given a Banach space $X$ and a measure space $(\Omega ,\mu )$, one says that a function
$$
f:\Omega \to X
$$
is Dunford integrable, or scalarly integrable if, for every $\varphi $ in the ...
- 2,233
3
votes
0
answers
65
views
Differential of smoothing operators
Following Gromov's article "Volume and bounded cohomology", given a riemannian manifold $M$, and given a covering $\tilde{M}$ with fundamental group $H \vartriangleleft \pi_1(M)$, we define ...
- 223
3
votes
0
answers
172
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Probability terminology
This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
- 22.5k
3
votes
0
answers
70
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Reference Request: Is every interval-valued probability measure consistent?
Short version: Does every interval-valued probability measure contain a conventional probability measure? I have a sense that this is a basic result about an obscure topic but I am having trouble ...
- 2,336