Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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2
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2answers
622 views

Question on Radon Nikodym Theorem

Let $\mu,\lambda$ and $\nu$ be $\sigma$-finite measures on $(X,M)$ such the $\nu\ll \mu$. Let $\mu= \nu + \lambda$. Then if $f$ is the Radon-Nikodym derivate of $\nu$ wrt $\mu$, we have $0\leq f\lt 1~\...
2
votes
1answer
228 views

continuity and measure

Let $ A, B \subseteq \mathbb {R} $ be Lebesgue measurable sets such that at least one of them has finite measure. Let $ f $ be the function defined by $$f (x) = m ((x + A) \cap B)$$ for each $ x ...
2
votes
1answer
1k views

Mutually Singular measures

c.f. Rudin's Real and Complex Analysis (Third Edition 1987) Chapter 6 Q9 Suppose that $\{g_n\}$ is a sequence of positive continuous functions on $I=[0,1]$, $\mu$ is a positive Borel measure on $I$, $...
2
votes
1answer
217 views

Can a measure with infinite integer moments be determined by its integer moments?

This question is motivated in part by an assumption I had to make in a write-up of this one. There are a number of well-known sufficient conditions for a measure to be determined by its moments, ...
2
votes
1answer
98 views

Show $f \in L^p$ implies there is a set $E$ and a function $g$ for which $f=f\chi_E+g$ with $m(E)<\infty$ and $|g|\le 1$

Let $f\in L^p$. Show that $f=f\chi_E+g$ where $m(E)<\infty$ and $|g|\le1$. Assume that $m$ is the Lebesgue measure on $\mathbb R$. Using Chebychev's inequality, I can find a set of finite measure $...
2
votes
1answer
293 views

BV functions and absolute continuity short inquiry

This comes as a complement to: Relation between total variation and absolute continuity; I was wondering if the following holds: Let $F$ be a function of bounded variation on $[a,b]$, then $\int_{a}^{...
2
votes
1answer
1k views

An equivalent definition of uniform integrability

Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each $\...
2
votes
1answer
120 views

Two norms of stochastic kernels

Let $(E,\mathscr E)$ be a measure space and $P:E \times\mathscr E\to [0,1]$ be a stochastic kernel - i.e. $$ P(x,A)\in [0,1] $$ for any $x\in E$ and $A\in \mathscr E$. On a set $b\mathscr E$ of ...
2
votes
1answer
126 views

Extending a set of complex Borel measures defined on subsets to the whole space

Continuing my work through Folland, trying to prove the following (Chapter 7 #22): Added: *Let $X$ be a locally compact Hausdorff space.* Let $\{f_\alpha\}_{\alpha\in A}$ be a subset of $C_0(X)$ ...
2
votes
1answer
266 views

Positive sequence of integrable functions

The question was: Given $\mu$ a positive measure in $(X, \Sigma)$ and $f_n, f:X\rightarrow [0,\infty)$ $\mu$-summable then show that if $\liminf f_n\geq f$ almost everywhere and $$\limsup_n \int_X ...
2
votes
2answers
673 views

Interpreting the Space of Square-Integrable Functions

I know that to construct the space $L^2( [-a,a) ) $, and to appreciate its richness, we need the machinery of lebesgue integration. However, I would like to work and talk about this space without ever ...
2
votes
1answer
149 views

Uniquely express a bounded linear functional on $C^k([0,1])$ as the sum of $k+1$ functionals?

Let $f$ be a function defined on a $J=[0,1]$, which is $k$-times differentiable on $J$; i.e. $f\in C^k(J)$. By the Riesz representation theorem we know that for any complex Radon measure $\mu$ on $J$...
2
votes
1answer
123 views

Exercise: Limits and Probability Measure

Let $\mu$ be a probability measure on $X$ (closed but unbounded), so that $\int_X \mu(dx) = 1$. Let functions $f_i:X \rightarrow \mathbb{R}_{\geq 0}$, $i = 1,2,...$, be Uniformly Integrable. Prove ...
2
votes
1answer
96 views

showing that $\left|\int_X f~d\mu\right| = \int_X |f|~d\mu \Leftrightarrow |f|=\beta f $ a.e

I have a problem I need help in solving. Suppose that $f\in L^1(\mu)$. I would like to show that $\left|\int_X f~d\mu\right| = \int_X |f|~d\mu$ if and only if $\exists$ a constant $\beta$ such ...
2
votes
1answer
163 views

measure preserving transformations on the ternary Cantor set

I'm working on a project where I define a new sort of measure defined as the following: $$m_\phi(E)=m(\phi^{-1}(E))\;,$$ where $E \subset C$ for the ternary Cantor set $C$. Mind you that $$\phi: \text{...
2
votes
1answer
199 views

An approach to Borel-Cantelli for the $l^p$

Let $\mu$ be a non-negative measure and $\{E_{k}\}$ a sequence such that $\sum \mu(E_k)^p<\infty $ then show that $F=\lim \ \sup E_{k}=\cap_{k=1}^{\infty}\cup_{n\geq k}E_n$ has $\mu$ measure zero.
2
votes
1answer
485 views

A question about the Lebesgue-Stieltjes measure of the Cantor function

This question is a follow-up to the post "Calculating a Lebesgue integral involving the Cantor Function." Let $\varphi: [0,1] \rightarrow [0,1]$ be the Cantor (ternary) function, and let $m_\varphi$ ...
2
votes
1answer
236 views

Weak convergence of measures and convergence of “almost densities”

Let $f_i$ be a sequence of smooth functions on $S^2$ such that the measures $\mu_i=f_i \;d\mathrm{vol}_{S^2}$ converge weakly to $d\mathrm{vol}_{S^2}$. Now suppose $\epsilon_i$ is a sequence going to ...
2
votes
1answer
324 views

Understanding no free lunch theorem

From Wikipedia: $Y^X$ is the set of all objective functions $f$:$X$→$Y$, where $X$ is a finite solution space and $Y$ is a finite poset. The set of all permutations of $X$ is $J$. A random ...
2
votes
1answer
130 views

Kolmogorov’s example of a measurable function not (generally) differentiable

In [1, page 7], the author says. Kolmogorov showed that if the function $$f(x) = \sum_{n=1}^{\infty} \frac{\cos 3^n x}{3^n}$$ has a finite or infinite generalized derivative on a set of positive ...
2
votes
1answer
526 views

A consequence of the Fubini-Tonelli theorem?

Tonelli-Fubini Theorem. Let $(\mathbb{X},\mathscr{X},\mu)$ and $(\mathbb{Y},\mathscr{Y},\nu)$ be probability spaces and let $\mathscr{Z}$ be the $\sigma$-field product i.e. the $\sigma$-field ...
2
votes
1answer
1k views

Generated Sigma-algebra example

I know that a sigma algebra generated by a subset a, $\sigma(a) = \{\emptyset,a,a^c,E\}$. But what about $\sigma({a,b})$? Would it be $\{\emptyset,a,a^c, b, b^c,E\}$?
2
votes
3answers
280 views

reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
2
votes
1answer
162 views

Existence of a structure-preserving mapping between two spaces?

I have some questions, but not sure if they are meaningful: Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$? Suppose $X$ and $Y$ are ...
2
votes
1answer
405 views

from almost everywhere convergence to uniform convergence

Suppose $\mu$ is sigma-finite measure on a space $X$, and $ f_{n}$ converge to $f$ almost everywhere. Show that there exists measurable sets $E_{n}$ where $\mu ( \cap E_{n}^{c}) = 0$ and $f_{n}$ ...
2
votes
1answer
519 views

How to construct an infinite family of cantor-like sets with certain properties

Q. Construct infinitely many disjoint sets $A_1, A_2,... \subset R$, each of which is a union of suitable symmetric Cantor sets, such that for every interval I and every $k=1,2,...$ the intersection $...
2
votes
1answer
276 views

R as a union of a zero measure set and a meager set

Let $ \left\{ {r_i } \right\}_{i = 1}^\infty = \mathbb{Q}$ an enumeration of $\mathbb{Q}$. Let $ J_{n,i} = \left( {r_i - \frac{1} {{2^{n + i} }},r_i + \frac{1} {{2^{n + i} }}} \right) $ $ ...
2
votes
2answers
252 views

How to show that $\frac{f}{g}$ is measurable

Here is my attempt to show that $\frac{f}{g}~,g\neq 0$ is a measurable function, if $f$ and $g$ are measurable function. I'd be happy if someone could look if it's okay. Since $fg$ is measurable, ...
2
votes
1answer
284 views

Separability of the set of positive measures

Let $X$ be a locally compact separable & metrizable space, and $M^{+}(X)$ its space of positive measures (i.e. positive linear forms on the space of continuous functions on $X$, continuous on each ...
2
votes
1answer
107 views

Jordan Measures without $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$?

I am trying to prove that Jordan measures satisfy with the following properties $A, B \subset \mathbb R$ and $d(A) = \sup( \{ d(x,y) | x,y \in A \} ) < \infty$, similarly for $B$: $$\bar{\mu} (A) \...
2
votes
1answer
281 views

An example of Markov-Feller chain with some properties

Let $X$ be a Polish space and $C(X)$ denote the space of all bounded and continuous functions on $X$. We consider a Markov chain $(\xi_n)_{n\geq 0}$ with transition probability $P:X\times \mathcal{B}...
2
votes
1answer
136 views

A sequence of functions $f_n \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$

Consider a sequence of functions $\{f_n \}\in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ , convergent to $f$ in $L^1(\mathbb{R})$ and to $g$ in $L^2(\mathbb{R})$. Prove that $f=g$ a.e. What I understood ...
2
votes
1answer
275 views

An exercise for weak convergence

Recently, I found an exercise in Hunter's Applied Analysis(last page in the link), which may be closely related to the question I raised two months ago. Consider heat flow in a rod with rapidly ...
2
votes
1answer
98 views

Random variable problem

Define a discrete random variable. Let $(Ω, A, P )$ be a probability space with $Ω = \{1,2,3,4,5,6\}$ and $F = \{Φ, \{1,3,5\}, \{2,4,6\}, Ω\}$. Define functions $X$, $Y$, $Z$ on $Ω$ as $X(k)= k$, $...
2
votes
1answer
258 views

Show that this function is not continuous except on a set of measure zero

Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals Let $$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$ where $$\chi_{(0,1]} = \left\{\begin{array}{ll} 1&...
2
votes
1answer
29 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
2
votes
0answers
20 views

Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ ...
2
votes
0answers
29 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
2
votes
1answer
37 views

Determining Class of a general Borel measure

Let $(X, \mathcal{T})$ be a topological space, and $\Sigma = \Sigma(\mathcal{T})$ the $\sigma$-algebra of Borel sets (that is, the $\sigma$-algebra generated by $\mathcal{T}$). In Real Analysis and ...
2
votes
0answers
37 views

If $F$ is a closed subset of $[0, 1]$, then how do I see that there exists a finite measure on $[0, 1]$ whose support is $F$? [closed]

If $X$ is a metric space, $\mathcal{B}$ is the Borel $\sigma$-algebra, and $\mu$ is a measure on $(X, \mathcal{B})$, then the support of $\mu$ is the smallest closed set $F$ such that $\mu(F^\text{c}) ...
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
2
votes
1answer
24 views

Sequence of nonnegative $f_n$ tending to $0$ pointwise where $\int f_n \to 0$, but there's no integrable function where $f_n \le g$ for all $n$?

What is an example of a sequence of nonnegative functions $f_n$ tending to $0$ pointwise such that $\int f_n \to 0$, but there is no integrable function such that $f_n \le g$ for all $n$?
2
votes
0answers
31 views

What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
2
votes
0answers
7 views

Can we approximate integrable functions by finite-valued upper,lower semicontinuous functions?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$ and $f\in L^1(\mu)$ be real-valued and $\epsilon>0$. Then, by Vitali-Caratheodory theorem, there exist upper ...
2
votes
0answers
29 views

Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
2
votes
0answers
34 views

Prove that $f$ is measurable.

Let $u$ be a subharmonic function on the disk $\Delta$ centered at the origin with radius $\rho$. $u$ is not identically $-\infty$. Define $f:\Delta\times [0,2\pi]\rightarrow [-\infty, \...
2
votes
0answers
30 views

Why $\lim_{x\to \infty }Mf(x)=0$ where $Mf$ is the Hardy-Littlewood maximal function.

In a course it is written that since $Mf(x)$ decay to zero at infinity, the measure or $\left\{x\in\mathbb R^n\mid\left|Mf(x)\right|>\lambda\right\}$ is finite. I was looking for such a result on ...
2
votes
0answers
26 views

If $\mu(E)\geqslant 0$ is it true that $E\in \mathfrak{M}$?

Suppose $(X,\mathfrak{M},\mu)$ be a mesure space. Let $E$ such that $\mu(E)\geqslant 0$. Can we conclude that $E\in \mathfrak{M}$? I think YES because $\mu$ is the set function with domain $\mathfrak{...
2
votes
0answers
47 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
2
votes
0answers
5 views

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...