Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,071 questions
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Zeros of functions involving polynomials and square roots
Say, you have a function from $C^n \to C$ which is built by adding and composing polynomials and square root functions (e.g., $f(x_1,x_2,x_3)=\sqrt{x_1^2+1}+\sqrt{x_2}+2\sqrt{x_3}$). Is there an easy ...
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Linear functionals and continuous functions on open intervals
Let $Q$ be an open interval of ${\mathtt R}$ and $E$ be the space of continuous and bounded functions in $Q\to \mathtt{R}$.
I call $E^*$ the set of linear functionals over $E$ and $E_+^*$ the subset ...
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Does pushforward preserve outer regularity?
(ZF + Countable Choice)
Let $\langle A,\mathcal{S} \hspace{.02 in} \rangle$ and $\langle B,\mathcal{T} \hspace{.06 in} \rangle$ be second-countable Hausdorff spaces.
Let $\Sigma$ be a sigma-algebra ...
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Follow up question on the measure of the difference between a partial selector and a selector...
This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...
In Kharazishvili's "Nonmeasurable Sets and ...
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Difference between a partial selector and a selector...
In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:
There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.
The proof is as follows:
...
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Uniform inverse-measure-preserving function for some class of measures
Working in the cantor space $2^\omega$.
Giving two measurable spaces $(2^\omega, T, \mu)$ and $(2^\omega, T, \nu)$ an inverse-measure-preserving function $f:2^\omega \rightarrow 2^\omega$ is such that ...
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Is an integrable map from a measure space to a Banach space always measurable?
Is every integrable mapping defined in a general measure space to a Banach space measurable?
The answer is yes if it is function (real valued). The answer is yes if it is a mapping into a Banach ...
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basic measure theory question - measure on the natural numbers [closed]
I am looking for a succinct way to describe a subset of the natural numbers which has ``measure zero" in the following sense: Let X_1 \subset X_2 \subset .... be any strictly nesting sequence of ...
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Weak convergence of measures on non-metrizable spaces
(ZF + Countable Choice)
Let $\langle X,\mathcal{T} \hspace{.06 in} \rangle$ be a second-countable Hausdorff space. Let $\mu$ be a Borel measure on $X$.
Let $\langle I,\leq_I \rangle$ be a directed ...
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Is the 2d gauge integral equivalent to the Lebesgue integral for nonnegative functions?
Let $f$ be a function from $[0,1]\times [0,1]$ to $\mathbb{R}$.
Definition:
2dgauge$\displaystyle\int f \; = \; I$
$\Leftrightarrow$
For all neighborhoods $U$ of $I$, there exists a function $\...
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Non-measurable sets and Determinacy...
Assume AC. Suppose $X$ is a subset of the irrationals (Baire Space) for which neither player has a winning strategy (i.e. the game $G(\omega, X)$ is not determined). Is $X$ non-measurable in the ...
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Expanding Measurable Sets
Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that
$$
\|f(x)-f(y)\| \;\geq\; \|x-y\|
$$
for all $x,y \in S$. Does it ...
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Is the sum of 2 Lebesgue measurable sets measurable?
Is the sum of two measurable set measurable? I think it is not...
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The Vitali Covering Theorem and use of closed balls
Concerning the Vitali Covering Theorem, what is the significance of closed balls in the hypothesis? In particular wouldn't open balls also work?
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Measures on infinite dimensional Banach spaces
Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
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derivation and measure
Given n vectors $w_1,\dots, w_n$ in $R^d$ we consider the functional $\phi$ defined on polynomials by $\phi(p)= \partial_{w_1}\dots \partial_{w_n}p(0)$ where $\partial_{w}$ is the directional ...
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To what extent can the following zero-one laws be relaxed?
I am interested in what circumstances various zero-one laws in probability theory can be relaxed. In particular, independence is a very important factor in such laws.
1) Borel-Cantelli Lemma: Let $...
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Exceptional Set in Egoroff's Theorem
I'll use the version of this question I posted on Stakexchange to replace the former version. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $...
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exotic compact group
Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...
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Universally measurable sets and weak topology
After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...
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Polar Coordinates and Borel sets.
Maybe this is just a stupid question, but I have tried very hard and get no luck. Here is the question:
Let $S_{k-1}$ be the unit sphere in $R^k$, i.e., the set of all $u\in R^k$ whose distance from ...
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Probability measure product space
Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces, and let $m$ be the product measure. Let $f:X \times Y \rightarrow [0,\infty )$ be a $B \otimes C $ measurable function, $1 < p < \infty$, ...
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Lebesgue measure of the graph of a function [closed]
Let $f: R^n \rightarrow R^m$ be any function.
Will the graph of f always have Lebesgue measure zero ?
1) I could prove that this is true if f is continuous.
2) I suspect it is true if f is ...
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Measure of infinite intersection of sets
Given a closed bounded set $X \subset \mathbb{R}^3$ and two curves $\gamma_1$ and $\gamma_2$ in the group of orientation preserving isometries of $\mathbb{R}^3$. Define the sets $X_1$ and $X_2$ as ...
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Measurable function is Baire class 2 almost everywhere
Let $X$ be a polish space (separable completely metrizable topological space). Let $m$ be a probability measure on $X$ and $f:X \rightarrow \mathbb{R}$ a measureable function. I want to show that $f$ ...
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Support of Probability Measures on Separable Metric Spaces
Let $X$ be a separable metric space and $p$ a probability measure on the Borel Sets of $X$.
Denote $S_p$ the support of $p$, i.e. the set of points which have positive measure for any ball around ...
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$\Delta_{2}^{1}$-hard set?
Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...
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What does progressively measurable actually entail?
There is a definition that has always left nagging questions in my mind. To set it up, let $(\Omega, \cal{A}, (\cal{F}_t)_{t\geq 0}, P)$ be a filtered probability space. From Comets & Meyre's ...
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If $G$ is amenable, when $G\times G$ is amenable ?
I am not specialist on Topological Group Theory, I apologize if this is a trivial question.
Question. If $G_1=G_2$ are amenable topological groups what additional hypothesis we have to consider on ...
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Question on measure theory [closed]
Here is the fact in measure theory:
FACT : Let $E$ be a Lebesgue measurable subset of $\mathbb{R}^n$. Almost every $x\in E$ satisfies $\lim\limits_{m(B)\to 0,~x\in B}\frac{m(B\cap E)}{m(B)}=1$ i.e. ...
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Is there a "disjoint union" sigma algebra?
I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets $\{A_i\...
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Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
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Sigma-algebras on Banach Spaces.
I am very interested in counterexamples when cylindrical sigma-algebra is not equal to the borel. I read
link text and
link text.
Especially interessted in l-infinity space. I've read Talgat and ...
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A question about regular signed or complex Borel measure under LRN decomposition
Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$, m is the Lebesgue measure on the class of Borel sets $\mathcal B_{\mathbb R^n}$ and the Lebesgue-Radon-Nikodym ...
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Is there a good approximating polygon for every smooth set?
Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is ...
- 3,270
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Are there non-measurable sets with smooth boundary?
I learned analysis a while ago, so let me define what I want. Suppose we have a set whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that this set is (...
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Jordan measurability of the level sets
Let $A$ be a compact subset of $R^n$ and $d_S(\bullet, A)$ be the
signed distance function of $A$. Namely, $d_S(p,A) =
d\left({p,\partial A} \right)$ for p in A, and $d_S(p,A) =
-d\left({p,\partial A}...
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Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...
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Probability Measures and Cardinality > c
Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be appreciated.)...
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Deterministic coupling of probability measures with a constrained support condition given a random coupling.
The following question has come up while working on my senior thesis: Assume $\mu$ and $\nu$ are regular probability measures on $\mathbf{R}^n$. We are also given a coupling $\gamma$ of $\mu$ and $\nu$...
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Non Separability of the the Loeb Space
Let $X_i$, $i=1,2\ldots$ be finite sets of integers such that $|X_i|\rightarrow \infty$. Let $X$ be an ultraproduct of $X_i$. Let $\mu$ the Loeb measure associated with the the normalized counting ...
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487
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On generalisation of Aizenman-Higuchi Theorem
Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$.
For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ ...
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Is total variation continuous?
Given a sequence of signed measures $<\nu_j>$, if it happens that $\nu=\sum\limits_{j = 1}^\infty \nu_j$ is still a valid signed measure (then it can be proved that each partial sum $\nu_n=\sum\...
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Codimension of Measurable Sets
I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...
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G-delta of measure 0 containig the rationals.
It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...
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Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
- 4,952
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Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
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Convergence of sets
Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the non-...
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Which categorical (coproduct-like) operation captures integration of measures?
Suppose we have a measure space $(X,a)$, a measurable space $Y$ and for every $x\in X$ we have a measure $b_x$ on $Y$. Suppose that $(Z,c)$ is a measure space such that as a measurable space $Z=X\...
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Measurability of sets of pairs
Suppose $(X,\mathcal{B},\mu)$ is a measure space, and let $B\subseteq X^2$ be an arbitrary set.
1) Is there a nice characterization of the circumstances under which there is a $\sigma$-algebra $\...
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