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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.

2
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0answers
139 views

3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.: (i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators; ...
0
votes
0answers
112 views

Quantitative Lusin’s theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in $...
1
vote
0answers
71 views

If $\text P\left[X_2\in B_2\mid X_1\right]=\kappa(X_1,B_2)$ a.s. for all $B_2$, can we select a common null-set over all $B_2$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E_i,\mathcal E_i)$ be a measurable space $X_1:\Omega\to E_1$ $X_2:\Omega\to E_2$ be $(\mathcal A,\mathcal E_2)$-measurable $\kappa$ ...
3
votes
1answer
94 views

Prove that $\mu \left(\left\{t\in X\,;\;\sum_{i=1}^d|\phi_i(t)|^2>r \right\}\right)=0$

Let $(X,\mu)$ be a measure space and $\phi=(\phi_1,\cdots,\phi_d)\in L^{\infty}(X)$. Let $$r=\max\left\{\sum_{i=1}^d|z_i|^2; (z_1,\cdots,z_d)\in \mathcal{C}(\phi)\right\},$$ where $\mathcal{C}(\phi)$...
4
votes
1answer
363 views

(non) separability of the power set

Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L_1$ pseudometric $\Delta$ on $2^X$, via $\...
0
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0answers
62 views

How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?

Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...
0
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0answers
148 views

How do we introduce a signed finite measure on the space of curves confined into the box $[0,1]^{n}$?

Given $\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$, consider the equivalence relation: \begin{align*} & \alpha_{1} \sim \alpha_{2} \Leftrightarrow \int_{0}^{1}...
5
votes
1answer
234 views

Variant of Sierpiński's result on non-atomic measures

Sierpiński's theorem states that nonatomic probability measures take a continuum of values. What if I assume that $\mu$ is a countably additive probability measure on $(X,2^X)$ and further that $\mu(\{...
4
votes
1answer
203 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
109 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
183 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
votes
0answers
60 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
87 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
825 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
125 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
115 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
votes
1answer
71 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
92 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
155 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: ...
1
vote
0answers
60 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
128 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 \...
7
votes
1answer
252 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
71 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
189 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
76 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
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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
61 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
192 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
44 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
90 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
243 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
116 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
97 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 \...
6
votes
2answers
191 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
125 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
100 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
378 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
94 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
412 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
106 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 $...
3
votes
1answer
235 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 $\boldsymbol{b}$ ...
1
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0answers
84 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
374 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
199 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
37 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 ...
11
votes
2answers
392 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 ...
4
votes
0answers
127 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
141 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
63 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
156 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 $\alpha_1,\dots,\alpha_k$. ...