Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,073 questions
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$\nu$ is a Dirac delta
Let $X$ be an locally compact Hausdorff space and $m$ a positive regular Borel probability measure where $m(Y)$ is 0 or 1 for any Borel set of $X$. Does it necessarily follow that $m$ is a Dirac delta?...
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Reference for a simple fact in measure theory (semi-algebras)
What is the textbook where the following simple fact can be found:
A measure defined on a semi-algebra S can be extended to a sigma-algebra generated by S.
In the texbooks that I have looked into ...
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Metrizability of the space of probability measures endowed with the topology of setwise convergence
Let $X$ be a separable completely metrizable space, let $\mathscr{B}(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathscr{P}(X)$ denote the space of all probability measures on $(X, \...
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The density one properties of $\mathcal H^{N-1}$-rectifiable set
Let $S\subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set. Then we know that there exist countably many Lipschitz $N-1$-graphs $\Gamma_i\subset \mathbb R^N$ such that
$$
\mathcal H^{N-1}\...
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Product of limit $\sigma$-algebras
Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...
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Can a nowhere differentiable function preserve measurability?
I want to know whether a continuous nowhere differentiable function $f: \mathbf{R} \to \mathbf{R}$ can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know ...
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Converge of measures [closed]
Good night, we have the following:
Let $(X,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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Is arbitrary union of closed balls in $\mathbb{R}^n$ Lebesgue measurable?
Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb{R}^n$ Lebesgue measurable? If so, is it a Borel set?
@George
I still have two questions concerning your sketch of ...
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Generalizing Integration by parts for general bounded continous measure
Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...
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When is the graph of a function a dense set?
Let $f: \mathbb R \to \mathbb R$ be any function. When is the graph of $f$ dense in $\mathbb R^2$?
The only examples I know for this are for non-measurable functions, but is that a necessary condition?...
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Extension of a function from almost everywhere to everywhere
The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...
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Formula for an integration on $\mathbb{Q} \cap [0,1]$
In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:
$$ \...
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An inequality for $L^p$-functions [closed]
I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + \int\...
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Measure of the Attractor of Critical Points of a Manifold
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
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Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
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List of Bernoulli chaotic systems
Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...
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Definitions of Hilbert Bundles
I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
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Extreme couplings
Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...
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Set of General Linear Position with Nonzero Measure
I came to the following question, but I don't have quite a good idea how to approach.
Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position?
I believe that,...
- 3,574
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Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
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Visualizing ANOVA Decomposition [closed]
Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...
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Bivariate integration with the range of one variable shrinking to a point
Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration
$$
\...
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surprisingly difficult filtration problem
I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
CommunityBot
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Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions
Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line ...
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Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
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$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$
I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
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Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...
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Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]
I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb R}xd\...
- 128k
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A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
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1
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Integration against Borel measures on compact Hausdorff spaces
I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...
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Are finitely generated amenable groups positively finitely generated?
Let $G$ be a finitely generated amenable group.
Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability?
Being more formal, note that $G^n$ is ...
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A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$
I asked this at math.stackexchange, but nobody answered.
Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\...
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Non-completeness of the Borel-Lebesgue measure and countable choice
Is it possible to prove the non-completeness of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) without the full axiom of choice, but still with Countable Choice ?...
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Does every (generalized?) Markov chain admit transition probabilities?
To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$ ...
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Pointwise convergence of a sequence of approximate limits of BV functions
So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
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Invariant probability on a unit ball of a Banach space
Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
- 11.6k
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Is there a name for this metric on a Borel sets
Consider a finite measure space $(X,\Sigma,\mu)$.
Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by
$$d(\sigma_1,\sigma_2) = \mu \left\{ (\sigma_1^c \cap \sigma_2) \cup (\sigma_1 \cap \...
- 1,590
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Restrictions on spectral measure
Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...
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Countably generated $\sigma$-algebra
Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space?
I assume not, so here is a more ...
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Measurable sets of probability measures $\{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}$
Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the ...
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Convex combinations of Bernoulli Measures
How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...
- 1,658
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364
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Bounds on Wasserstein (Kantorovich) distance
Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...
- 1,655
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measure of an image under an argmax function
I am trying to find any techniques to analyze the measure of an image of a set under an argmax function.
For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
2
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1
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Do we have independence if we let the indices of the events increase?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
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Interchanging integrals and continuous linear forms in RKHS
I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
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297
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Approximating characteristic functions by cutting the real axis into smaller and smaller pieces
Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
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Radon-Nikodym derivative as a limit of ratios
This question is related to Radon-Nikodym derivatives as limits of ratios.
Let $F$, $G$ be sigma-finite measures (or at least probability measures) on $\mathbb{R}$ such that $F \ll G$.
The theorem ...
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How to show that there's a continuous function separating convex sets of Radon measures?
First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
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Compactness of cadlag martingales w.r.t. to the point-wise topology
Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...
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Uniform convergence of long geodesic to Liouville measure
Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...
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