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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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
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47
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Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
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113
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The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
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125
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Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
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134
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Are there any books or literature on norms over measure space?
Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying ...
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78
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Different measurability of Hilbert-space valued random variable
My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...
- 503
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101
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Approximation arguments
I am a student reading about Luo and Sarnak's paper and I have trouble understanding the conclusion.
In the paper this theorem is proved for a continuous function of compact support $\psi$:
$$\...
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146
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Remainder-balancedness of primes
Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...
- 48.9k
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130
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Approximate range of Radon-Nikodym derivative in a dynamical system
Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...
- 307
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Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
- 657
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Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 835
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118
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A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
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If $\kappa$ is a Markov kernel with density $p$, does it generally hold $p(x,z)=\int p(x,y)p(y,z)\:{\rm d}y$?
Let $(E,\mathcal E)$ be a measurable space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$. Assume that $$\kappa(x,B)=\int_Bp(x,y)\:\lambda({\rm d}y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$...
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If $X$ is a right-continuous process, is $t\mapsto\operatorname E\left[X_\tau\mid\tau=t\right]$ right-continuous as well?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(X_t)_{t\in[0,\:\infty]}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$;
$\tau$ be an $[0,\infty]$-valued random ...
- 167
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48
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Sets measurable in every affine subspace
Take a non-measurable subset $S\subseteq [-1,1]$ and subtract $S\times \{0\}$ from the unit disk $B$ in $\mathbb{R}^2$. The set $X=B\setminus (S\times \{0\})$ is measurable by 2-D Lebesgue measure ...
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Reference request for the dual space of the Bochner Space $L^1(\Omega ; X) $
Let $\Omega\subset\mathbb{R}^N$ be open and let $X$ be a Banach Space. Let $L^1(\Omega ; X)$ denote the space of all strongly measurable (sometimes also referred to as Bochner Measurable) functions, $...
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 463
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66
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Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 463
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390
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Monotone convergence theorem for increasing net of positive functions
Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question.
(Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
- 1,879
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233
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Setwise convergence of induced measure converging weakly
I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.
Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $...
- 63
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21
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Estimatives for elliptic systems involving the laplacian
Considering the problem
\begin{equation}
\left\{
\begin{array}[c]{11}
\Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\
\Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\
\end{...
- 101
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75
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Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
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116
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Is there a proper term for a "continuum-convex" set?
Let $X$ be a Banach space, and for any compactly supported Borel probability measure $\mathbb{P}$ on $X$, define the mean $\mu_\mathbb{P}$ by $\mu_\mathbb{P}=\int_X x \, \mathbb{P}(dx)$.
I want to say ...
- 708
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Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
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50
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How to define finitely additive transition probabilities
I'm wondering how does one define finitely additive transition probabilities. Let $\mathcal X_1$ and $\mathcal X_2$ denote the Borel $\sigma$ algebra of two topological spaces $X_1$ and $X_2$. Suppose ...
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70
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Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
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78
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Evaluating a limit at a discontinuity of a monotone rearrangment (distribution function)
I have a question that occurred to me and has been bothering me, because maybe graphically it seems obvious but I don't know how to get there. It has to do with the distribution function and monotone ...
- 143
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186
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Critical exponent in Sobolev scale and singular continuous measure
Recently I started the following discussion about Sobolev spaces where I was interested in finding the maximal $s \in \mathbb{R}$ such that given function/distribution belongs to the Sobolev space $H^...
- 9,330
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For $T\subseteq\mathbb R$, when is $\mathbb Q + T$ measurable? Of positive measure?
If we take one element from each coset of $\mathbb Q$ in $\mathbb R$, we obtain a Vitali set, which is unmeasurable. What if instead of a choice of representatives, we take the union of a subset of ...
- 623
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153
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Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
- 59
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221
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Measure concentrated on the $\omega$-limit set
Let $(X,F)$ be a dynamical system with $X$ a compact metric space and $F: X\to X$ continuous. By $\omega$-limit set of a subset $A\subset X$ I mean:
$$\omega(A):= \bigcap_{n=0}^\infty \left(\overline{\...
- 4,459
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403
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Measurability of centered Hardy-Littlewood maximal function with doubling measure
Let $(X, d, \mu)$ be a metric space with doubling measure $\mu$ a.e. every open ball has finite and positive measure $\mu$ and there exists $C>0$ such that
$$\mu(B(x,2r)) \le C\mu(B(x,r))$$
for ...
- 137
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80
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On the weak convergence of sub-probability measures
Denote by $M(\mathbb R)$ the collection of sub-probability measures. Let $(\mu_n)_{n\ge 1}\subset M(\mathbb R)$ and $\mu\in M(\mathbb R)$. Do we have the equivalence of the following claims :
$\mu_n$ ...
user128095
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133
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Convoluted Cantor-like measure which has a continuous component [duplicate]
Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
\sum_{k\ge 1}3^{-k}X_k
$$...
- 1,352
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77
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Norm on the space of finite signed Borel measures over a compact metric space
Let $S$ be a compact metric space. Let $\Delta^\ast(S)$ be the set of all finite signed Borel measures defined on $S$, and $\Delta(S)$ be the set of all Borel probability measures defined on $S$.
My ...
- 157
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148
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Classifying non atomic singular measures up to topological conjugacy
Write $\mathcal S$ for the set of probability measures on $[0, 1]$ that are non atomic and singular with respect to Lebesgue measure.
Two measures $\mu$ and $\nu$ in $\mathcal S$ are said to be ...
- 6,205
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235
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Lebesgue measure of a neighbourhood of a curve
Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary).
For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
- 1,759
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88
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Sequence of open sets converge in characteristic function to an open set?
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
- 1,759
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86
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A non trivial example of a Gaussian semi-Markov process?
Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov ...
- 143
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87
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How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?
I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...
- 537
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92
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Can $\ell_1(E)$ be embedd into the dual of continuous function space?
Let $E$ be a compact seperable space, for example $E=[0,1]\subset\mathbb{R}$. Denote by
$$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\...
- 131
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59
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Examples of strongly continuous measure-valued functions
Let $X$ be a compact geodesic metric space and let $P_p(X)$ be the set of all finite Borel measure on X with finite $p^{th}$ moment. We equip $P_p(X)$ with the total variation topology metric. What ...
- 5,405
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71
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Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
- 5,405
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151
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Definition of conditional expectation for singleton
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. Furthermore, let $X, Y$ be two random variables from our probability ...
- 33
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139
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How many moments determine a normal distribution?
I know that a Gaussian distribuion is determined by its moments. I was wondering if there is a result of the form:
if we know that the first thousand moments of a random variable are Gaussian, then is ...
- 3,062
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302
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Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
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1
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88
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Recurrence results for an "on average" measure preserving transformation
I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$
$$
\lim_{n\rightarrow \...
- 1,087
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97
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Wigner semicircle law and random measures
tl;dr: the proof of the Wigner semicircle law seems to confuse measures with random measures. I do not understand why. Scroll down until 'QUESTION' if you are fine with the theoretical stuff.
T. Tao ...
- 91
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150
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Define the convolution root of probability measures on a measurable group
Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$.
Remember that a probability ...
- 167
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85
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
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