Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2,897
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Comparing two $\sigma$-algebras
Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.
Q. For which ...
4
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2
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432
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Lifting back the induced invariant measure / general version of Kac's formula for occupation times
Let $T$ be a conservative measure preserving (non-invertible!) transformation of a measure space $(X, \mathscr{F}, m)$ with infinite measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup_{k=0}^\...
6
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1
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643
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On the failure of extending a probability measure on uncountable $\Omega$
It is a well known fact that if $(\Omega, \mathcal{F}, P)$ is a probability triple and $\{A_i : i < k\}$ is a finite collection subsets of $\Omega$, then there is a $P' \supset P$ and $\mathcal{F'} ...
8
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Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?
I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
2
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1
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351
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Daniell integral vs. Borel measure
Let $X$ be a locally compact, Hausdorff, topological space and denote by $\mathcal B_+(X)$ the collection of all Borel-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals).
...
6
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2
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588
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Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
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0
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112
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Constructing a measure from an integral
A well known Theorem by Riesz says that every continuous linear functional on the space $C(X)$ of all complex valued continuous functions on the compact space $X$, is given as the integral against a ...
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370
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Comparing two $\sigma$-algebras on $B(\ell^1)$
Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow
$$w-\lim T_i=T \Longleftrightarrow \...
5
votes
2
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401
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Do a Hausdorff space and its associated completely regular space have the same Borel subsets?
Let $(X,T)$ be a Hausdorff topological space. Let $C_b(X)$ be its algebra of continuous bounded functions. Let $T'$ be the initial topology on $X$ given by $C_b(X)$. It is known that $T=T'$ if and ...
6
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3
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422
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Point-wise limit of finite valued functions
Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?
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119
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Convergence of $L^p$ of approximation
Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients
$$\alpha_{ij}= \frac{1}{\...
6
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2
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468
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Continuity of disintegrations
Suppose that $\pi:Y\to X$ is a continuous surjection from one compact metric space to another. Given a regular probability measure $\mu$ on $Y$ with pushforward measure $\nu:=\pi^*\mu$, it is known ...
0
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0
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171
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Regularity of level sets of Sobolev derivatives
I am interested in the regularity of the sets
$$U_{\lambda}:=\{x: |\nabla^k u(x)|> \lambda \}$$
for a function $u\in W^{k,p}(R^d)$ with compact support. We can choose a lower semicontinuous ...
4
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321
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Topological field theories and their path integrals
Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten ...
2
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1
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232
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A measure of noncompactness by a convex function
Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\...
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Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
2
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1
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303
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Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
1
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0
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114
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Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$
Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}...
1
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1
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163
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Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
2
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1
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102
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Measurability of the product on particular topological vector spaces
Let $X$ be a topological vector space. Let us say that $X$ has property P if there exists a sequence of closed subsets $\{X_n\}$ such that
1- $X=\bigcup X_n$
2- The relative topology is both ...
2
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0
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Distribution of a multivariate continuous process determined by that of linear combination of its coordinates?
To keep the question short: Let $C([0,1], \mathbb{R}^d)$, $d \geq 2$ be the space of all $\mathbb{R}^d$-valued continuous processes. $X$ and $Y$ are two $C([0,1],\mathbb{R}^d)$-valued random variates, ...
1
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1
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298
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Bochner measurable; continuous operator
It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
6
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Measurable functions with non measurable image
I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable.
This is motivated by the question Is measure preserving function almost surjective?, ...
0
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1
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133
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Change of variables for double integral [closed]
Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to ...
3
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0
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Does every non-locally compact metric space admit a violation of Lebesgue's theorem?
From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...
1
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0
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1k
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Sigma algebra of stochastic process
A stochastic process is a collection $(X_t)_{t\in T}$ of random variables from a prob. space $(\Omega,\mathcal{F},P)$ to some measurable space $(E,\mathcal{E})$. Now, in order to understand the whole ...
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137
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Weak$^*$ topology on Bochner $L^p$-spaces
Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
2
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1
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555
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Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
4
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2
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413
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Non-measurability of time integral of non-jointly measurable process
I'm teaching a seminar on probability theory and I want to motivate why joint measurability of a stochastic process is important. The following seems to be the canonical counterexample for a process ...
3
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1
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288
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Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
In Theorem 2.7 in the following notes, we seem to assume the following statement.
Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
2
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1
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229
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Absolute continuity of infinite product of probability measures
Let $(A_i,\mathcal{B}_i,\mu_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\...
3
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1
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202
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A particular measure of noncompactness?
I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...
9
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2
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735
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Functions that are approximately differentiable a.e
The classical definition of an approximately differentiable function is as follows:
Definition.
Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
3
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2
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259
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The disintegration of the convolution of two probability measures
Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
3
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1
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77
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Convergence of some object depending on functions with compact support
Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...
16
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2
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972
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Is measure preserving function almost surjective?
Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one?
This question is motivated by the following observation. If ...
0
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1
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154
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Positive upper asymptotic density and equidistribution
Let $B=\{b_n: n\geq 1\}$ be a set of positive integer numbers with positive upper asymptotic density and let $\alpha$ be a real irrational number.
Is it true that $\{b_n \alpha\}$ is equidistributed ...
2
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1
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192
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Non-uniqueness in Krylov-Bogoliubov theorem
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a ...
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0
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66
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Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
2
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2
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Identity map between $L^2(\mu)$ and $L^2(\mu_{\rm sf})$
Let $(X,\Sigma,\mu)$ be a measure space.
The semi-finite version of $\mu$ on $(X,\Sigma)$ is denoted $\mu_{\rm sf}$ and given by
$$
\mu_{\rm sf}(E) = \sup\{\mu(A) \mid A \subseteq E \text{ measurable,...
2
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1
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Explicitly representing a random variable in terms of indicator functions
Motivation:
I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula.
I want to prove the change of variable formula (you ...
1
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1
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192
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Is this function measurable?
Let $(B, \Sigma_B)$ and $(C, \Sigma_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$.
Given $Y\in \Sigma_{B\times C}$, I would like to use the following function:
$$
f:...
1
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1
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66
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Measurability of kernel on generating set
Suppose $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ are measurable spaces such that $\Sigma_Y$ is generated by a set $B$.
Suppose $k : X \times \Sigma_Y \to [0, 1]$ has the property that $k(x, -)$ is a (...
4
votes
2
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141
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Invariant function for Koopman operator of measure-class preserving tranformation
Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (...
0
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1
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121
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Approximation of a measure on $\mathbb{R}^d$
Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix:
$$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$
I'm trying to prove the existence of a probability measure ...
7
votes
2
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851
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If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.?
If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that
...
3
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0
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81
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Is Lebesgue measure a measurable mapping under Fell topology?
Suppose $\mathcal{F}$ is the space of closed set on $\mathbb{R}$ equipped with the Fell topology.
Question: Is Lebesgue measure a Borel measurable mapping from $\mathcal{F}$ to the extended real ...
2
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1
answer
138
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Measurability of $T \to \Pi_{ker T}$ w.r.t. SOT
I came across the following technical question, to which I could not - after some time of thinking - find an answer:
Let $\mathcal{U},\mathcal{H}$ be two real (in general infinite dimensional) ...
0
votes
1
answer
52
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Binarily universal members of $[0,1]$
Let $r\in[0,1]$. We look at the binary represenation of $r$ and say that $r$ is binarily universal if every finite binary string appears in at least one place in the binary representation of $r$. Let $...
3
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1
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173
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A conjecture: Given $nD$ finite measures that bisects
Given $nD$ finite measures $µ_1, . . ., µ_{nD}$ in $\mathbb{R}^n$
does there exists a set of at most D hyperplanes that bisect each of the measures?
How far is this problem explained?
That is ...