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Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

3
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1answer
167 views

Does measurability of cardinal $\kappa$ imply measurability of $2^\kappa$?

A cardinal $\kappa$ is real-valued measurable if there is a $\kappa$-additive probability measure on $2^\kappa$ which vanishes on singletons. The existence of measurable $\kappa$ is independent of ZFC....
0
votes
1answer
53 views

Total variation norm estimate

I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\...
1
vote
0answers
159 views

Random variables over large measurable cardinals

This question assumes the existence of a large real-valued measurable cardinal. Let $X$ be an uncountable set and $(X,2^X)$ equipped with a non-atomic probability distribution $P$. Additionally, let $...
0
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0answers
48 views

convergence of a fraction containing Lebesgue measures and perimeters

Sorry for this basic question posted on MO, I as well asked here https://math.stackexchange.com/questions/3063202/convergence-of-a-fraction-containing-lebesgue-measures-and-perimeters Let $\epsilon&...
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0answers
55 views

When does measurability of sections imply measurability in product $\sigma$-algebra?

Let $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ be measurable spaces and let $\mathcal{Z}$ be the product $\sigma$-algebra on $X\times Y$. Suppose $A\subset X\times Y$ is such that its $X$- and $Y$-...
4
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0answers
82 views

How to think about dual space of a certain space of Lipschitz functions

Consider the following Banach space (for concreteness): $$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$ where $$ \bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
19
votes
1answer
779 views

Fubini without CH

In Real and Complex Analysis, Rudin gives an example (due to Sierpinski) of a function $f:[0,1]^2\to[0,1]$ separately Lebesgue-measurable in each argument, such that $$ \int_0^1 dx\int_0^1f(x,y)\,dy \...
2
votes
1answer
105 views

Difference quotient for functions of bounded variation

Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation. We have that the following holds $$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
0
votes
1answer
110 views

Given these conditions, can a function be defined that is well defined a.e.?

I have two functions, and I want to combine them to define a certain function. Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a....
0
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1answer
64 views

Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$

Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...
0
votes
1answer
87 views

Are these conditions enough to ensure joint measurability?

Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?
8
votes
1answer
153 views

On the existence of a particular type of finite measure on $\mathbb N$

Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
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0answers
56 views

On the convergence of an integral of Hardy's maximal function

Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function. Assume that $$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ ...
3
votes
1answer
116 views

Invariant integration on principal bundles

Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...
8
votes
1answer
241 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 ...
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0answers
67 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
187 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
73 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 ...
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0answers
45 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
57 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
174 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 ...
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0answers
38 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
1answer
81 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
169 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
107 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
90 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
180 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
113 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
90 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
326 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
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0answers
86 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
405 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
103 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 $...
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0answers
27 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
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0answers
120 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
223 views

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 $\boldsymbol{b}$ ...
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0answers
79 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
361 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 ...
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0answers
34 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 ...
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0answers
33 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
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2answers
359 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 ...
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0answers
50 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 \...
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0answers
44 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 $\phi\mapsto (\mu/\nu)^r\phi :$...
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0answers
59 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
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0answers
124 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
140 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
57 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
134 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 $\alpha_1,\dots,\alpha_k$. ...
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0answers
115 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 ...