Skip to main content

Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

Filter by
Sorted by
Tagged with
-1 votes
1 answer
138 views

On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components

Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent. Now, I was wondering ...
Learning math's user avatar
-1 votes
1 answer
653 views

Convergence in the narrow topology of measures and strongly converge for signed measures

We say that a sequence $(\mu_{n})$ of measures in $M_{b}(Q)$ converges tightly (or, equivalently, in the narrow topology of measures) to a measure $\mu$ in $M_{b}(Q)$ if $$\lim_{n\to\infty}\int_{Q}\...
M.A's user avatar
  • 43
-1 votes
1 answer
83 views

Convergence in mean and convergence in distribution

Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that $$ 0< ...
Wenguang Zhao's user avatar
-1 votes
1 answer
259 views

Absolute continuity of probabilities on Polish spaces and open sets. [closed]

On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
Polite's user avatar
  • 41
-1 votes
1 answer
477 views

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 ...
user693's user avatar
  • 135
-1 votes
1 answer
129 views

(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes

Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda ...
Wenguang Zhao's user avatar
-1 votes
1 answer
988 views

Random variable as an integral of an indicator function

This answer says that if $X$ is a random variable and $X_+ = \mathrm{max}(0, X)$, then $X_+ = \int_0^\infty I_{\{X > x\}}\mathrm{d}x$. I'd like to know how to derive this starting with $A \in \...
johnsmith's user avatar
  • 115
-1 votes
1 answer
76 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
di sun's user avatar
  • 1
-1 votes
1 answer
696 views

Can singular measures be viewed as vanishing distributions? (Answer No!)

Hello, Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
Anand's user avatar
  • 1,649
-2 votes
1 answer
880 views

a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
mike's user avatar
  • 603
-2 votes
1 answer
146 views

a measure convolution equation

My question is: Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
mostafa's user avatar
  • 367
-2 votes
1 answer
347 views

Forms of multivariate CLT [closed]

I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
wanderflo's user avatar
-2 votes
1 answer
108 views

If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent

Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$. If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
0xbadf00d's user avatar
  • 167
-2 votes
0 answers
64 views

A Problem using Limits of Sequences of Functions

Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
hunter's user avatar
  • 1
-3 votes
1 answer
537 views

Measure Theories with a different convention to $\infty\cdot 0 =0$ [closed]

As we all know in a first course in measure theory we define a symbol $\infty$ to satisfy $\infty \cdot 0=0$, but there are more two possible choices for a convention as someone has shown to me; one ...
Alan's user avatar
  • 1,594
-3 votes
2 answers
195 views

Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]

Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
user530909's user avatar
-3 votes
1 answer
276 views

Why surreal numbers cannot be extended further in this way using measure approach?

Basically, a lebesgue measure of dimension $n$ of a set of the same dimension $n$ is $n$-volume, $\lambda_n(S)$. If the dimension of a set is greater than the dimension of the measure, the measure is ...
Anixx's user avatar
  • 10.1k
-3 votes
2 answers
156 views

Getting almost certainty from uncountably many low-probability events

Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
ABIM's user avatar
  • 5,405
-3 votes
0 answers
65 views

Exercise generalizing (related to) Hölder's inequality

I came across this exercise and feel absolutely stuck: Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
HZA's user avatar
  • 1
-4 votes
1 answer
66 views

Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$? [closed]

Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that $$ \sup_n\int_{E}|f_n|d\mu<+\infty. $$ Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
Made's user avatar
  • 115
-5 votes
1 answer
270 views

Calculus based on pdf [closed]

Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
Marius S.L.'s user avatar

1
58 59 60 61
62