Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
1,549 questions
7
votes
1answer
216 views
What's so special about the Orthonormal base $\{e_n\}$ of $L^2[0,1]$, where $e_n(x)=e^{2\pi i nx }$?
Let $f \in L^2([0,1])$ . Then Carleson's Theorem states that
$$\lim_{N\to \infty} \sum_{|n|<N} \langle f,e_n\rangle e_n(x)=f(x),\quad\text{a.e. } x\in[0,1],$$
where $\{e_n\}$ is the Orthonormal ...
1
vote
0answers
43 views
Why do weak and L metric topology for measures coincide?
Let $(X,d)$ be a complete metric space,
$$M^1 := \{\mu: \mu \mbox{ is a Borel regular measures having bounded support and } \mu(X) = 1\},$$
and $$BC(X) := \{f : f:X\rightarrow \mathbb{R} \mbox{ is ...
9
votes
1answer
174 views
Rothberger property for finite covers
Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...
1
vote
2answers
70 views
Positive part of “outer sums” of measures
Here is a question about decomposition of measures in singular parts and in positive and negative parts.
$\newcommand{\RR}{\mathbb{R}}$
Let $\Omega_{1/2}$ be compact subsets of $\RR^d$ equipped ...
1
vote
0answers
44 views
Matching Stochastic Flows
Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...
0
votes
1answer
56 views
Evolution of a density under the doubling angle map
Let $\mu$ be a probability measure on $I=[0,1]$, absolutely continuous with respect to Lebesgue measure. Denote by $T$ the "doubling angle map" on $I$, where $T(x)=2x \text{ mod }1$. Is it true, in ...
3
votes
2answers
128 views
Example of measure for some algebra over N
$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...
1
vote
0answers
35 views
Extension of a result about measurable, additive functionals
Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
1
vote
0answers
58 views
An example of a measurable random process with non-measurable integral
Let $ \xi _t(\omega), t\in[0,\infty)$, be a random process and let $ \xi _t(\omega)\in \{\mathfrak F_t\}$ be some filtration. Even if $ \xi _t(\omega) $ is $ \mathfrak F_t $ measurable then $\int_0^...
0
votes
2answers
142 views
Uncountable infimum of measurable functions
I know that in general it does not hold, but there exists a positive result under some conditions for which the following claim holds?
Claim: given an uncountable set of measurable functions $f_i : U ...
1
vote
0answers
98 views
How to show that this function is continuous (Geometric Measure Theory)
I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
2
votes
1answer
84 views
Measurability of C([0,1]) for the completion of the Wiener measure
Consider the completion $(\mathbb{R}^{[0,1]}, \mathcal{B}, \mu)$ of the Wiener measure on $\mathbb{R}^{[0,1]}$ (with the cylinder set $\sigma$-algebra).
Is the following true :
$C([0,1])\in \...
5
votes
2answers
172 views
Extension of a von Neumann algebra by a von Neumann algebra
I asked this question at MSE now I repeat it at MO:
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0\to A\to C\to B\...
2
votes
2answers
106 views
Weak convergence of measures and a sequence of closed sets
Assume that we have a metric space $(A,\rho)$ and a sequence of probabilistic Borel measures $\mu_{n}$ on $A$ that converges weakly to the probabilistic Borel measure $\mu$. Assume also that one is ...
4
votes
1answer
86 views
Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
5
votes
1answer
324 views
Base zero-dimensional spaces
Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
-2
votes
0answers
51 views
Estimate on the difference between the measure of the sublevels of two functions in terms of their $L^1$ distance
Fix $R\gg 1$. How can I estimate the difference between the Lebesgue measures $$\mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): f(x)>0\} - \mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): g(x)>0\...
2
votes
0answers
78 views
Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?
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\...
7
votes
3answers
400 views
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\...
4
votes
0answers
98 views
Geometric interpretation for the Lebesgue-Radon-Nikodym Theorem
Discussing with some friends, the following question arose:
If $\nu$ is a signed measure, $\mu$ is a positive measure, and they're both $\sigma$-finite, then we may write $\nu = \lambda+\rho$, where $...
0
votes
0answers
25 views
Is there a Markov family that does not have an associated semigroup?
After reading some references, I found that many probabilists are cautious about whether or not a Markov family is associated with a semigroup, while many others assume its existence for granted.
So ...
2
votes
0answers
115 views
Infintely iterated and functional integration in constructive math
Looking for references on constructive derivations of (elements of) functional integration -- in particular, those used in the classical construction of the Wiener measure.
It seems such ...
2
votes
1answer
216 views
+50
Meaning of Alberti rank-one theorem
Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
1
vote
0answers
78 views
size of local strong stable manifold is measurable
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
13
votes
1answer
353 views
reverse mathematics of the Lebesgue measurability of analytic sets
Can the fact that all analytic sets are Lebesgue measurable be proven in $Z_2$, or in some weak subsystem such as $\Pi^1_1\text{-CA}_0$? Conversely, can certain set existence axioms be derived from ...
8
votes
1answer
195 views
Is a measurable solution continuous?
Let $f: \mathbb R\to \mathbb R$ be a Borel measurable function. Suppose that for each $q\in \mathbb Q$, the function $f(q+x)-f(x)$ is continuous on $\mathbb R$. Is it true that there is a continuous ...
1
vote
0answers
33 views
If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?
Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...
0
votes
0answers
32 views
Predictable Process Question (Da Prato & Zabcyzk 2014)
I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $\Phi$ is adapted and ...
11
votes
2answers
345 views
Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?
Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...
1
vote
0answers
49 views
If $P \ll Q$, are the regular conditional probabilities a.s. absolutely continuous?
Let $P$ and $Q$ be probabilities on $(\Omega, \mathcal{A})$, and let $\mathcal{F}$ be a sigma-subalgebra of $\mathcal{A}$. Assume $P \ll Q$. Assume that $P(\cdot \mid \mathcal{F})$ and $Q(\cdot \mid \...
0
votes
0answers
42 views
References for formal powers of measures
In Information geometry, Ay et al. define the space of formal $r$th powers* of signed measures as the limit $\mathcal S^r(\Omega) = \injlim L^{1/r}(\Omega, \mu)$ of maps $\phi\mapsto (\mu/\nu)^r\phi :$...
0
votes
0answers
57 views
Separable measures on compact groups
Let us say that a (signed, finite) measure $\mu$ is separable if $L_1(|\mu|)$ is a separable Banach space.
EDIT: Suppose that $G$ is a locally compact group such that each measure on $G$ is ...
4
votes
0answers
121 views
Gelfand spectrum as a measure space
Given a Lebesgue probability measure space $(X,m)$ (say, just the unit interval with the Lebesgue measure on it), let $A$ be a closed subalgebra of the real $L^\infty(X,m)$. Then one can realize the ...
0
votes
1answer
137 views
Support of a regular measure Reg
Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...
3
votes
0answers
56 views
Integration on a family of differential forms
Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
3
votes
1answer
130 views
Minkowski sum of polytopes from their facet normals and volumes
By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...
1
vote
0answers
103 views
Egorov's and Lusin's Theorem in the space with infinite measure
Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite.
On the measurable space whose measure is infinite, does there ...
1
vote
1answer
40 views
Monotonicity given an implicit function containing a Measure integral
The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers!
Knowing that $$f(\...
0
votes
1answer
123 views
Regularity in Orlicz spaces for the Poisson equation
I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007)
$$\Delta u=f \quad \quad \quad \quad (1)$$
Lemma 2: There is a constant $N_1 >1$ so that ...
3
votes
2answers
221 views
Effect of perturbing the atoms of a measure on the Wasserstein distance
Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
1
vote
1answer
154 views
continuity entropy with respect gibbs measures
Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...
3
votes
0answers
104 views
Completely I-non-measurable unions in Polish spaces
Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
2
votes
1answer
99 views
Measure preserving coordinates of $S^2$ from $[0,1]^2$
Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.
Question ...
0
votes
0answers
34 views
Weak* convergence in a dual Banach lattice vs norm convergence of moduli
Let $E$ be a dual Banach lattice, that is, $E = E_*^*$ for some Banach lattice $E_*$ (I have $M(X)=C(X)^*$ specifically in mind for a compact space $X$).
Suppose that $(x_n)$ is a weak*-null ...
6
votes
1answer
175 views
Weak*-convergence of signed measures
Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...
3
votes
1answer
95 views
Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$
For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some ...
2
votes
1answer
211 views
Is there a finitely additive measure on R which is not sigma-additive?
Consider the usual measurable space of real number $( \mathbb{R}, \mathcal{B}(\mathbb{R}))$.
My question is:
Is there an application $\mu$ on $\mathcal{B}( \mathbb{R}) \rightarrow [0,+\infty]$ such ...
8
votes
2answers
446 views
Is taking the positive part of a measure a continuous operation?
Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.
Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure ...
3
votes
1answer
61 views
Echange of Infimum Integral with Pointwise Infimum
Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
2
votes
0answers
79 views
Inequality concerning BV norm
Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...
