Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,894
questions
9
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1
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A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
1
vote
0
answers
50
views
Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(E, |\...
0
votes
0
answers
42
views
Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
2
votes
2
answers
71
views
Reference for Wiener type measure on $C(T)$ when $T$ is open
I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
0
votes
0
answers
109
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The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
1
vote
0
answers
29
views
Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
2
votes
1
answer
163
views
Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product
I've been trying to understand various questions to do with sigma algebras on uncountable product spaces.
Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
3
votes
0
answers
131
views
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 ...
1
vote
0
answers
59
views
Is the projection of an universally measurable set again universally measurable?
Let $(X,\mathcal{A})$ be a measurable space and $(Y,\mathcal{B}(Y))$ be a polish space together with the Borel-$\sigma$-Algebra.
There is a Theorem that states: The projection $\pi_X(B)$ of every ...
0
votes
0
answers
83
views
Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Is $F: X \to L^p_{\text{loc}} (Y), x \mapsto f(x, \cdot)$ Bochner measurable?
Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
1
vote
1
answer
166
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
3
votes
2
answers
138
views
Mollifying a measure without changing its marginals
Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ ...
1
vote
0
answers
77
views
Is this metric on the space of $\mu$-measurable functions complete?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
0
votes
2
answers
84
views
How to construct this sequence that converges a.e. in product measure and that has a very particular form?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
3
votes
3
answers
468
views
Solving interval problems without outer measure
Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ?
Problem 1
If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
0
votes
1
answer
138
views
For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
0
votes
2
answers
119
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
1
vote
1
answer
137
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
3
votes
2
answers
315
views
A general inequality for KL divergence of functions of variables
The question concerns a very general setting and a very general inequality about KL divergence. I'm writing this thread to verify whether my intuition is correct.
Let $E_1, E_2$ be two measurable ...
1
vote
1
answer
226
views
What is the measure of two sets which partition the reals into subsets of positive measure?
This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.
(In ...
4
votes
1
answer
138
views
Is the set of clopen subsets Borel in the Effros Borel space?
Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
1
vote
0
answers
75
views
Borel structure/sets coming from strong operator topology vs norm topology
Let $X, Y$ be Banach spaces. Moreover, let $\mathcal{L}(X,Y)$ be the space of bounded linear operators equipped with the standard operator norm topology, and $\mathcal{L}_{\mathrm{s}}(X,Y)$ the same ...
1
vote
1
answer
81
views
Approximating a family of measurable functions
Let $X$ be a set of $N>0$ elements (with the counting measure) and consider a family of measurable functions $f_i:X\to [0,1]$, for $i\in \mathbb N$.
Any function $f_i$ can be seen as a point in the ...
0
votes
0
answers
28
views
Concerning the conversion of an essential supremum to a pointwise estimate
In the following paper :
Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian, Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3747-...
0
votes
1
answer
142
views
Billingsley convergence of probability measures - inequality used in Theorem 2
On Page 8, Billingsley defines $f(x)=(1-\rho(x,F)/\epsilon)^{+}$ where $\rho(x,F)$ is the metric distance from the set $F$. He then states $|f(x)-f(y)|\leq \rho(x,y)/\epsilon$ and goes on to use this ...
1
vote
0
answers
138
views
Polish spaces and analytic sets
Can we conclude that an analytic subset $A$ of a Polish space $X$ is also Polish?
Let $\mathcal{M}(R^d)$ denotes the family of Borel probability measures on $R^d$ equipped with the Lévy-Prokhorov ...
1
vote
1
answer
188
views
What is convergence in distribution of random variables taking values in a non-metrizable product space?
Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...
5
votes
1
answer
356
views
Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?
Suppose $\lambda^{*}$ is the Lebesgue outer measure.
Question:
Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where:
The range of $f$ is $\mathbb{R}$
For all real $x_1,x_2,y_1,y_2$, where $...
2
votes
1
answer
147
views
Finding an explicit & bijective function that satisfies the following properties?
Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$.
Question:
Does there exist an explicit ...
2
votes
0
answers
119
views
Measure algebra for a family of probability measures
Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and
\begin{align}
B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \}
\end{align}
the ...
0
votes
1
answer
120
views
An integral Minkowski inequality for the quasi-Banach case?
The so called integral Minkowski inequality claims that for suitable positive functions $f$ defined on the product of two measure spaces $ (X\times Y, \mu \times \nu) $ and $p\geq 1$ we have that
$$ \...
4
votes
0
answers
161
views
Finding balls with big measure
Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under ...
2
votes
0
answers
45
views
$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
5
votes
1
answer
163
views
Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions
Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
3
votes
2
answers
248
views
Representing measurable map to compact space as a continuous map
Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space
$$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...
9
votes
3
answers
764
views
How to prove that the Lebesgue $\sigma$-Algebra is not countably generated?
How to prove that there can't exist a countable set $\{A_1,A_2,\dots\}\subset \mathcal{L}(\mathbb{R})$ (where $\mathcal{L}(\mathbb{R})$ denotes the family of all Lebesgue measurable sets) such that $\...
9
votes
1
answer
1k
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Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?
So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) ...
6
votes
0
answers
260
views
Is there a uniform version of Lebesgue's differentiation theorem?
Let $\mu$ be a finite measure on $\mathbb R$ and $f,g : \mathbb R \to \mathbb R_{\geq 0}$ two measurable maps such that $\int_{x\in\mathbb R} f(x)\ \mu(dx) \leq 1$ and that $g(x) \leq 1$ for all $x$. ...
4
votes
0
answers
100
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
0
votes
1
answer
194
views
Density of "Fibonacci friends"
Let $F$ be the set of all integers $n>1$ such that in the Fibonacci sequence modulo $n$, the value $0$ occurs infinitely often. What is the value of $\lim\sup_{n\to\infty}\frac{|F\cap\{0,\ldots,n\}|...
7
votes
0
answers
128
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
0
votes
0
answers
117
views
Are there any books or literature on norms over measure space?
Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying ...
2
votes
0
answers
118
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
0
votes
0
answers
68
views
Using programming to measure the uniformity of measurable subsets of the unit square?
This is a follow up to this post using this answer:
Let $S:=[0,1]^2$ be the unit square. "Partition" $S$ naturally into four congruent squares $S_{1,j}$ (with side length $1/2$ each), where ...
5
votes
1
answer
312
views
Quantitative Lebesgue density theorem
Let $A \subset [0, 1]$ be a measurable set, and $\mathbf 1_A$ its indicator function, viewed as a function on $\mathbb R$. Define for each $\delta > 0$, the function $f_{A, \varepsilon}: \mathbb R \...
1
vote
1
answer
123
views
Integration of vector function against vector measure
Let $X,Y,Z$ be Banach spaces and let $m\,:\,X\times Y\to Z$ be a bilinear map such that $\|m(x,y)\|\leq C \|x\|\|y\|$ for some fixed constant $C$. Moreover, let $\mu$ be a Borell vector measure on $\...
1
vote
0
answers
112
views
Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
1
vote
1
answer
138
views
Convergence rate of a sequence of sets to a set-theoretic limit?
Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.
If we define a sequence of sets $\left(F_r\right)_{r\in\mathbb{N}}$ with a set theoretic limit of $A$; how do we define the rate at which $\...
0
votes
0
answers
78
views
Purely non-atomic measure on the Gromov boundary of a finitely generated free group
In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...
5
votes
1
answer
185
views
If every point is a Lebesgue point of $f$, does $f$ satisfy the intermediate value property?
Let $f: \mathbb R \to \mathbb R$ be a locally integrable measurable function.
We say $f$ satisfies the intermediate value property if given any $a, b\in \mathbb R$ with $a < b$, whenever $u \in \...