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

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21 views

Non-regular measure can be represented by a regular measure

Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular). Then the linear map $L : C_c(X) \to \Bbb C$ ...
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1answer
73 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
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0answers
19 views

If a set has finite outer measure, why can you find a countable number of sets that cover it?

Let $μ: S \to [0, \infty]$ be a set function defined on a collection $S$ of subsets of a set $X$ and $\bar{\mu}: M \to [0, \infty]$ the Caratheodory measure induced by $μ$. Let $E$ be a subset of ...
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0answers
19 views

Why are the positive measure and negative measure induced by the Hahn Decomposition mutually singular?

The following statement describes the Hahn decomposition and claims that the induced positive measure and negative measure are mutually singular. Why is that the case? On a separate note, what are ...
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1answer
38 views

Let $A$ be a Lebesgue measurable set and let $0\leq b\leq\mu(A)$. Show there is measurable $B\subset A$ with $\mu(B)=b$

Let $A$ be a Lebesgue measurable set and let $0\leq b\leq\mu(A)$. Show there is measurable $B\subset A$ with $\mu(B)=b$. This is a qual problem. I tried approaching this with inner regularity. ...
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0answers
47 views

Lim Sup and Measurability of one Random Variable with respect to Another

Here, there is a common proposition in probability theory : Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that ...
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1answer
11 views

Is the map $\omega\mapsto (X(\omega),Y(\omega))$ measurable with respect to $\sigma(X,Y)$?

Definitions: We have a measure space $(\Omega,\sigma(X,Y),\mu)$ where $X,Y$ are maps from $\Omega$ to some measure space $(S,\Sigma,m)$. Here $\sigma(X,Y)$ is the smallest sigma algebra that makes ...
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0answers
20 views

linear dependence and measure theory

Let $(\Omega,\Sigma,\mu)$ be an measure space. Further let $E:=L^p(\Omega,\Sigma,\mu)$ where $\dim E \ge 2$. First I had to show, that if there are $[f],[g] \in E$, so that $\forall \alpha,\beta \in ...
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2answers
47 views

A real analysis problem on the integral inequality.

For fixed $ 0 < \alpha < \beta $, is there a positive constant $C_0$, depending only on $\alpha$ and $\beta$, such that for any bounded measurable function $ \varphi : \mathbb{R}^+\rightarrow [0 ...
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1answer
25 views

Prove that $f^*(x)\geq \frac{c}{\|x\|^n}$ if $f$ integrable.

Let $f:\mathbb R^n\longrightarrow \mathbb R$ a non-zera integrable function. Set $$f^*(x)\geq \sup_{B\ni x}\frac{1}{|B|}\int_B f(y)\mathrm d y,$$ where the supremum is taken over all the ball that ...
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2answers
64 views

Real Analysis, Folland Problem 1.3.15 Measures

Given a measure $\mu$ on $(X,M)$, define $\mu_0$ on $M$ by $$\mu_0(E) = \sup\{\mu(F): F\subset E \ \text{and} \ \mu(F) < \infty\}$$ a.) $\mu_0$ is a semifinite measure. It is called the ...
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2answers
134 views

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
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0answers
46 views

Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu $$ for all compactly supported smooth ...
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0answers
19 views

Integrals as Signed Measures (and vice Versa)?

1. Can every integral (with respect to an integrable function) be written as a signed measure? And does the function’s decomposition into positive and negative parts align somehow with the ...
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1answer
67 views

How can I prove the following equation.

For $0\le{}x\le1$ and for $0\le{}y\le1$, $f(x, y)$ satisfies that, for each $x$, $f(x, y)$ is an integrable function of $y$ and $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
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2answers
29 views

Exterior measure does not satisfy additivity. [closed]

Exterior measure $m^*$ satisfy subadditivity. $$m^*(A\cup B) \leq m^*(A) + m^*(B).$$ But for disjoint sets $A$ and $B$ it may be that $m^*(A\cup B) < m^*(A)+m^*(B)$. So I learned that It is one ...
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1answer
36 views

Does there exist a curve with non zero area?

Does there exist a curve with a bounded infinitely diffrentiable derivative (i.e. has a minimum |curvature|) of hausdorf demension 2? Or even a diffrentiable curve of hausdorf demension 2?
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2answers
51 views

Find a subset of A such that its boundary does not have measure zero

Question Find a subset $A$ of $[0,1]$ such that $A=cl(intA)$ and yet $bd(A)$ does not have measure $0$. I don't know how to construct it. I think it should be closed set, cannot be empty by ...
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18 views

Can anyone help me understand one step in the proof of claim bounded Lebesgue integrable functions defined on a set of finite measure is measurable?

Here is the outline of the proof. Take $\phi_n$ to be a sequence of increasing simple functions approaching f. Since $\phi_n$ is simple and measurable, f is measurable. My question is where in the ...
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2answers
33 views

$ L^p $ and inequality

I am trying to solve the following problem in Measure Theory. I assume that I have to use Hölder's Inequality but I don't see how. Let $ E $ measurable, $m(E)<+\infty$, $1<p<+\infty$ and $ ...
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0answers
61 views

Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
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1answer
39 views

Problem 2.2 Real Analysis Folland

I want to solve this problem: If $f,g:X\to \overline{\mathbb R}$ are measurable and $c$ is any extended real value the function $$h(x)=\begin{cases}c& \text{if}\ f(x)+g(x)\ \text{is undefined}\\ ...
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1answer
29 views

What is the measure zero of uncountable set .

Recently I was reading Methods of Real Analysis by Goldberg and had the following question. 7.1 Corollary: Every countable subset of $\mathbb{R}$ has measure zero. How can we describe the ...
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0answers
29 views

The relationship between random variables, distribution functions and probability measures

Given a probability space $(\Omega,\mathcal{F},P)$, and a random variable $X\colon\Omega\to\Bbb{R}$, we can associate with it its distribution function $F\colon \Bbb{R}\to[0,1]$ defined as ...
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3answers
49 views

Real Analysis, Folland Problem 1.3.11 Measures

Background information - Let $X$ be a set well equipped with a $\sigma$-algebra $M$. A measure on $M$ (or on $(X,M)$ or on $X$ if $M$ is understood) is a function $\mu: M \rightarrow [0,\infty]$ such ...
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15 views

Modulus of Integral defines a measure?

Say you have an integrable function $f: \mathbb{R}^n \to \mathbb{R}$ which does change sign. Does the set function $$m(A):=\left| \int_A f \text{d} x\right|$$ define a either a measure or a ...
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1answer
21 views

On the interpretation of the limsup of a sequence of events

In the context of understanding Borel-Cantelli lemmas, I have come across the expression for a sequence of events $\{E_n\}$: $$\bigcap_{n=1}^\infty \bigcup_{k\geq n}^\infty E_k$$ following Wikipedia ...
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0answers
19 views

Measurability and integrability of set and function

My textbook said: Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure. if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given ...
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1answer
25 views

$f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. Find $\lim \int f_n d\lambda$.

Let $X = \mathbb R$, $\textbf{X} = \textbf{B}$ and $\lambda$ the Lebesgue measure on $\textbf{X}$. I have the following: $f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. I need to find the following: ...
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1answer
34 views

Sequence of Events - Basic understanding

I have an idea of the meaning of a sample space, and the events included in a sigma algebra. However, I am stuck in the definition of a sequence of events. My difficulty is in the fact that in a ...
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0answers
46 views

Find the Lebesgue measure of the following sets.

Find the Lebesgue measure of the following sets: i) A=$(\cup_{n=1}^\infty [2^n, 2^n + \frac{1}{2^n}))$ \ $\mathbb{Z}$ ii) B=$(\cup_{n=1}^\infty (n^n, n^n + \frac{1}{2^n}))$ $\cap$ $\mathbb{Q}$. For ...
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0answers
12 views

Given an event field, is there a random variable generating it? [duplicate]

In probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, for any event field $\mathcal{G}\subset\mathcal{F}$, there always exists a random variable $X$, such that $\sigma(X)=\mathcal{G}$? Is ...
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0answers
21 views

Conditional Distributions vs. Stochastic Processes

Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ...
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0answers
50 views

Sigma-algebra on $\Omega = \{1,2,3,4,5\}$ generated by $\epsilon = \{\{1,2,4,5\},\{2,3\}\}$

Let $\Omega = \{1,2,3,4,5\}$. Let $\epsilon = \{\{1,2,4,5\},\{2,3\}\}$. Find $\sigma$ ($\epsilon$), generated by ($\epsilon$), and justify answer. Could someone please give me some direction and ...
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2answers
26 views

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$.

Let $f_n = (\frac{1}{n})\chi_{[0,n]}$ and $f = 0$. Show that $(f_n)$ converges uniformly to $f$. I have never done an example of convergence of sequences that have characteristic (indicator) ...
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2answers
38 views

Real Analysis, Folland Problem 1.3.14 [duplicate]

If $\mu$ is semifinite measure and $\mu(E) = \infty$, for any $C > 0$ there exists $F\subset E$ with $C < \mu(F) < \infty$. Attempted proof - Suppose $E\in M$ with $\mu(E) = \infty$ then ...
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1answer
42 views

Can the unit interval be the disjoint union of countably many “super-dense” parts?

I'm curious about this question in the case where $f$ is not necessarily measurable. I think what it comes down to is this: Is there an $\varepsilon<1$ and a partition of $[0,1]$ in countably ...
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1answer
57 views

Decomposition of complex Radon measures

Suppose you have a complex Radon measure $\mu$, treated as a distribution. Then does every such Radon measure admit a decomposition of the form $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
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24 views

Is there a there a non intersecting mapping to unit square.

Is there a way to go from the fat cantor set to a half unit square in a non intersecting way using Hilberts curve? How would I go about constructing a non intersecting space filling curve of non zero ...
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1answer
48 views

Is every set of measure zero countable?

I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?
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1answer
23 views

Exercise 10.N of The elements of integration and Lebesgue measure Bartle's book

If $a_{mn}\ge 0$ for $m,n\in\mathbb{N}$, then $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}a_{mn}(\le +\infty).$
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15 views

Prove that $\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$ is measurable.

If I have two measurable functions $f:\mathbb R^p\to\mathbb R^m$ and $f:\mathbb R^q\to\mathbb R^n$, how can I prove that $$\phi:\mathbb R^{p+q}\to\mathbb R^{m+n}:(x,y)\mapsto (f(x),g(y))$$ is ...
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1answer
34 views

A question on atoms in measure theory.

The definition in Bogachev's book goes as follows: Now let $\mathcal{A}$ be a $\sigma$ algebra and let $\mu$ be a finite countably additive measure. 1.12.7 Definition. The set $A\in \mathcal{A}$ ...
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1answer
31 views

A finitely additive measure is a measure if and only if we have continuity from below

A finitely additive measure $\mu$ is a measure if and only if it is continuous from below. I want to know how I should proceed in proving this statement. My idea is to first assume we have a ...
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3answers
71 views

$f $ vanishes iff $f$ integrable.

Can someone give me a sketch for proving this: Let $f:\mathbb R\to\mathbb R$ be a monotone and therefore measurable function. Show that $f$ is integrable if and only if $f(x)=0$. Some hints would be ...
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1answer
60 views

Real Analysis Folland, Problem 1.3.8 Measures

If $(X,M,\mu)$ is a measure space and $\{E_j\}_{1}^{\infty}\subset M$, then $\mu(\liminf E_j) \leq \liminf \mu(E_j)$. Attempted proof - Let $(X,M,\mu)$ be a measure space and ...
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4answers
70 views

$\int_\Omega f d\mu = 0 $ if and only if $f(x)=0$ almost everywhere

can someone give me a hint on what kind of theorem/definition I should make use of to solve this? Let $(\Omega,\mathfrak A, \mu)$ be a measure space and $f:\Omega \to \mathbb R$ a non-negative ...
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1answer
49 views

$\int_{A} fdxdy=0 $ For every rectangle A with area 1. Then is it f=0 a.e? [closed]

Is it true that the function $f \colon \mathbb R^2 \to \mathbb R$ satisfying condition $\int_{A} f \,\textrm{d}x \,\textrm{d}y=0$ for every rectangle $A$ whose area is $1$ must be identically 0 ...
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1answer
27 views

What is the Difference Between a Version and a Modification of a Stochastic Process?

Under what circumstances would one say that: The stochastic process $X$ is a version of the stochastic process $Y$? Background: See here for a related but slightly different question on ...
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0answers
23 views

Can anyone explain one step in the proof of the Lebesgue theorem?

If the function f is monotone on the open interval (a, b), then it is differentiable almost everywhere on (a, b). Proof: Assume f is increasing. Furthermore, assume (a, b) is bounded. ...