Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,071 questions
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Location of nontrivial Gleason parts on the topological boundary of the polydisc - Was it described anywhere else besides in Bekken's PhD Thesis?
My PhD advisor and me need the exact description of the location of the non-trivial Gleason parts on the topological boundary of the polydisc $\mathbb{D}^n$.
It was described in Otto B. Bekken's PhD ...
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Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
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Sufficient condition for the graph of a measurable map to be measurable
Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.
If $(X,\Sigma_X)$ is a standard Borel space can we always ...
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Are measurable maps with countably separated image in a Banach space always strongly measurable?
Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
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Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Does the following integration method hold for regular enough functions $F:\mathbb{R}\to\mathbb{R}$?
\begin{align}
&\zeta(2)\sum_{\frac{a}{b}\in\mathbb{Q}_n} \frac{F(\log \frac{a}{b})}{\sqrt{abn}...
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Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous？
Is there an increasing function on
$[a, b]$ which is differentiable,
but not absolutely continuous？
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A reference for an equation of evolution for a probability measure
I assume that there exist a family of probability measures $(d\mu_{t})_{t\geq 0}$ over the circle $\mathbb{R}_{|2\pi\mathbb{Z}}$ satisfying the following equation of evolution: for every continuous ...
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Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
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A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
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Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
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Choosing a relative large density subsequence from a low density sequence
My question is somewhere in the interface of combinatorics, probability, and measure theory. It is quite ad-hoc, and I wonder if there is a counter example.
Consider for example the unit interval $[0,...
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When a null uncountable set can be image of some increasing function with discontinuities on a dense countable set
Consider the following result:
A: Let $f:D \to \mathbb R$ be an increasing function with discontinuities on a dense countable subset of $D$ such that the jump values sum to $\mu(D)$, where $D$ is a ...
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Fourier coeffients of Cantor measure
For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is
$$
\...
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Mappings that preserve local or global minimum
In the most general form, I'm interested in any non-trivial results of the following question.
Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
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Why should one still teach Riemann integration?
In the introduction to chapter VIII of Dieudonné's Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument:
Finally, the reader will ...
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Ratio of Gaussian measure over Euclidean balls
Let $\nu\in\mathcal P(\mathbf R^d)$ be the standard Gaussian distribution $\mathcal N(0,I_d)$.
Denote by $\mathscr B$ the class of Euclidean balls $B_r(x)$ (centered in $x\in\mathbf R^d$ with radius $...
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Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
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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 ...
CommunityBot
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Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
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When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?
This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
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Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
CommunityBot
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Hausdorff distance and Hausdorff measure of symmetric difference
Let $X_n$ be a sequence of $k$-dimensional piecewise smooth submanifolds of $\mathbb{R}^m$, converging in Hausdorff distance to a $k$-dimensional piecewise smooth submanifold $Y \subset \mathbb{R}^m$, ...
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Path-homotopy in Wasserstein space
Consider two vector fields $b_0,b_1\in C^2([0,1]\times\mathbb{R}^d;\mathbb{R}^d)$ and the solutions $\rho_0,\rho_1\in AC([0,1];\mathcal{P}_2(\mathbb{R}^d))$ to the associated Fokker-Planck equations
$$...
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Convolution of a bounded function and measures
Given a function $f\in L^\infty(\mathbb{R}^n)$ and a family of Radon measure $\mu_\alpha$, under what condition do we have $f*\mu_\alpha$ equi-continuous?
One condition I know is if $\mu_\alpha$ has a ...
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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
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Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
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Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
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When does an inner product on $C(K)$ come from integration?
Let $K$ be a compact Hausdorff space and suppose $\langle \cdot, \cdot \rangle$ is an inner product on $C(K)$ such that $\langle f, g \rangle \ge 0$ whenever $f(t),g(t)\ge 0$ for all $t \in K$. It ...
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Convergence of iterated conditional expectations
Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$.
Let $X$ be an ...
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Question about Bochner measurability
When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and $...
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Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
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Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
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Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
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4
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For what sets does the Lebesgue Differentiation Theorem hold in one dimension?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
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Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
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A question on the cardinality of sigma-algebra generated by $\aleph_0$ or $\aleph_1$ class
This question comes from notes to section 1.2 in page 40-41 of Folland's "real analysis: modern techniques and their applications", 2nd edition. At the end of this note, the author asserts ...
CommunityBot
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Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
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A question on Borel measurability
Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is an infinite Borel measure and $\mu$ is not $\sigma$-finite. Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \...
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Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations
Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
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Integrability in the product space can follow from a property of the Nemytskii operator?
Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
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Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
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85
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Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
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Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
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Integral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:
$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$
$f(\...
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1
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Gelfand–Yaglom–Perez Theorem for product space
In On a Gel'fand-Yaglom-Peres theorem for f-divergences, Gilardoni proved the Gelfand–Yaglom–Perez Theorem for general $f$-divergence, i.e. $f$-divergence between two probability measures $P$ and $Q$ ...
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Ramsey type property of the Lipschitz constant
The following problem was proposed by Pietro Majer as an extension of an earlier question of mine on Lipschitz functions.
For $f$ a Lipschitz function on $\mathbb R^n$, we denote by
$$\text{Lip}(f, U) ...
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106
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Measure theory for non-Hausdorff topological spaces
I m interested in measures on non-Hausdorff spaces but have not been able to find anything specific beyond the general standard measure theory.
Can anyone please point me to references that focus on ...
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Deriving an inequality for the integral of maximum indicator functions under measure-preserving transformations
Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $...
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2
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$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
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Functions whose product with every $L^1$ function is $L^1$
Let $\mu$ be a probability measure and $f$ a measurable function whose
product with any integrable function is integrable: $$
\int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$
...
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