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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Difference in probability distributions from two different kernels
Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P_x$ and $\tilde{\mathsf P}_x$ differ on the space of ...
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Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
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Random Functions and Transition Probabilities
Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability ...
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A name for Radon-Nikodym derivatives that are bound away from zero and infinity
Dear Mathoverflow,
I would like to know if the nomenclature of mathematics has a name for Radon-Nikodym derivatives that are bounded away from zero and infinity almost everywhere. As in for ...
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What's the use of a complete measure?
A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
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Applied Problems in Probability which can not be modelled on Polish spaces
Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable space?...
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Pointwise limit at Lebesgue's point
Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
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What relates to measure spaces as topological spaces relate to metric spaces ?
Has there been study of a generalization of measure spaces along the following or similar lines ?
Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
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3
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Bounded operators and axiom of choice
In the article below, it is shown that the proposition "Every linear operator defined on a whole Hilbert space is bounded" is consistent with the axioms of ZF + a weakened version of the axiom of ...
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Measures on Riemannian manifolds which are not induced by the volume form of some Riemannian metric
Let $M$ be a smooth oriented manifold. Does there exist a smooth measure $m$ on $M$ which is not induced by the volume form of some Riemannian metric $g$ on $M$? I would say that the set of volume ...
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On the set of infinite measures
My question is about the structure of the set of infinite Borel measures on compact metric spaces invariant with respect to a homeomorphism.
Let $T$ be a homeomorphism of a compact metric space $X$ ...
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Conditions for non-triviality of Caratheodory measure
This may be too vague to end up being useful, but:
Are there any (natural? reasonable?) conditions that can be imposed on an outer measure $\phi^*:{\mathcal P}(S)\to[0,\infty]$ to ensure that the $\...
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Caratheodory and Riesz
This is related to a previous question.
Carathéodory's construction assigns to each outer measure $$\phi^+:\mathcal{P}(S)\to[0,\infty]$$ a $\sigma$-algebra $\Sigma$ where the restriction $\phi$ of $\...
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Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
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Is the following statement true? $E[\xi U^{'}(\xi)] < +\infty$?
I encounter the following problem today. It seems a simple question.
Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:
(1) $U$ is concave, continuous, ...
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Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
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Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
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Atoms of a sequence of Sigma-algebras
I'm trying for some time now to prove or disprove the following conjecture to no avail:
Let $S$ be a set and let $(\Sigma _n)$
be a sequence of countably generated
$\sigma$-algebras on $S$ ...
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from affine matroid to measures
Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the ...
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A question about measures on groups
Let $G$ be a finitely generated (in my case also amenable) group and $f:G\to[0,1]$. Suppose that there is a finitely additive probability measure $\mu$ on $G\times G$ and a real number $L$ such that $\...
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Probabilities Measures
Consider then a convex subset of probability measures $A$ that is closed relative to the set of all probability measures. I'm wondering if $\forall \mu \in$ closure($A$), does there exists a positive ...
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Is Jordan outer measure finitely additive on positively separated sets in $\mathbb{R^n}$?
I am trying to argue that exterior measure has nice properties that Jordan outer measure doesn't have. One of them is finite additivity, but I can't find a simple way to show Jordan outer measure is ...
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Two standard probability spaces
Let $(X,\cal{A},\mu)$ be a standard (Lebesgue-Rokhlin) space with complete probabilistic measure (for example, $[0,1]$). Let $\cal{B}\supset \cal{A}$ be a wider then $\cal{A}$ $\sigma$-algebra on $X$, ...
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Probability that a Turing machine will nontrivially reduce a real
For a fixed Turing machine $\Phi_e$, what is the probability that it will reduce a given real to some less complex, yet still non-computable real?
More precisely: It is known that the set of reals ...
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1
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Atoms of regular Borel measure
Let $X$ be locally compact Hausdorff space. Let $\mu$ be a Borel measure on it which is finite on compact and outer regular with respect to open sets and inner regular with respect to compact sets. ...
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Composed function made Lebesgue integrable?
Let $p(x)$ be a probability density function on the unbounded set $X \subseteq \mathbb{R}^n$, so that $\int_X p(x) dx = 1$.
Let $F: X \rightarrow \mathbb{R}_{\geq 0}$ a measurable but non-integrable ...
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Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
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A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
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6
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2
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Are Vitali-type nonmeasurable sets determinate?
Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from ...
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Universally measurable sets and the perfect set property
Is it true that all universally measurable sets (say on $[0,1]$ ) have the perfect set property?
I am not an expert in this at all and the answer may be known, but I was not able to find it.
I ...
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Why are simple functions defined for positive coefficients (in measure theory) [duplicate]
Possible Duplicate:
Why is Lebesgue integration taught using positive and negative parts of functions?
Hey,
I am currently referring 'probability with martingales'. To develop lesbegue ...
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Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
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The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
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When are iterated limits of sets equal to a double limit?
Suppose $\mathcal{A}$ is a $\sigma$-algebra, and $A_{1,1},A_{1,2},...,A_{2,1},A_{2,2},... \in \mathcal{A}$ is a double sequence of measurable sets. Under what circumstances do we have the following?
...
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Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation
Let $X$ be a compact Hausdorff space. It is well-known that every homomorphism $F : \mathcal{C}(X) \to \mathbb{R}$ is the evaluation $f \mapsto f(x)$ at some point $x \in X$. The usual proof is not ...
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Do signed measures on sigma-rings always have a Hahn decomposition?
Let $X$ be a set.
Let $\mathcal{R}$ be a set of subsets of $X$ such that
$\{\} \in \mathcal{R}$
and
For all members $A$ and $B$ of $\mathcal{R}$, $\;\; (A\cup B)-(A\cap B) \; \in \; \mathcal{R} \;\;$...
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Measures that satisfy a 0/1 law
The setting is measure on $2^\omega$. That product (independent) measures obey a 0/1 law, i.e, that measurable tail sets all have measure 0 or 1, is well known. I've made some progress extending this ...
- 2,044
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Sections measure zero imply set is measure zero?
I have a subset $B\subset\mathbb{R}^n\times\mathbb{R}^m$ that I want to show has measure zero. I know that the sections $B^x = \{y : (x,y)\in B\}$ all have measure zero. I do not know if $B$ is ...
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What does it mean to say "almost always" ?
I have a set, $A$, of $m \times n$ matrices with certain properties and a subset $B$ of $A$. I would like to say that when randomly selecting such a matrix, I am "almost always" never in $B$. I can ...
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Derivative of the Lebesgue integral. Currents.
I consider a special kind of sets in $\mathbb{R}^n_+$ given by $G_t = $ {$x \in \mathbb{R}^n_+ \mid g(x) < t$}, where $\nabla g > 0$ entrywise. Let's consider an integral
$$
f(t) = \int\...
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Can we change the Lebesgue measure by forcing?
Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...
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Kolmogorov probability axioms without non-negativity condition
What is a minimal consistent modification of probability axioms to include negative values?
Is it enough to use a minimal modification of axioms obtained by
formal exclusion of non-negativity ...
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Quotients of Measurable Spaces?
Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ ...
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Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
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The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
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Property Sigma Algebra [closed]
Is the set { $ \cup_{i \in \mathbb{N}} C_{i} \times D_{i} : C_{i} \in \mathcal{L} \ , D_{i} \in \mathcal{B}^{n} \ $ } a sigma algebra on $\mathbb{R} \times \mathbb{R}^{n}$ ?
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surjective function from non-measurable sets
let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval ...
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Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
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Problem of moments. Some generalisation.
Let $g(x) \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ be homogeneous of order 1, concave and smooth function. Using entrywise product notation $x \circ p = (x_1 p_1, ..., x_n p_n)$ one can write
$$
\...
- 1,329
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2
answers
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On the generalisation of Bernstein's theorem on monotone functions
Bernstein's theorem states that for any completely monotone function $f$: $f \in C^{\infty}[0,+\infty)$, $(-1)^n f^{(n)}(t) \geqslant 0$ there is a finite Borel measure $\mu$ such that
$$ f(t) = \...
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