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

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3
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1answer
104 views

Measurability and almost sure convergence

I am having trouble understanding the measurability issues arising with almost sure / almost everywhere convergence. $X_n \rightarrow X$ a.s. if $\Pr \{ \lim X_n = X \} = 1$. Put differently, ...
1
vote
0answers
53 views

the existence of density preserving map

Let $X$ be a random varible defined on the probability space $(R^n,F; P)$, $Y$ be a random varible defined on the probability space $(R^m,F; P)(m<n)$, does there exist a map $f:X\rightarrow Y$ such ...
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0answers
92 views

Prove the Lebesgue measure of a triangle is $\frac{1}{2}|(A-C)\wedge (B-C)|$

I'm new to this site so tell me if this question is written wrong. Given a triangle with vertex's A,B,C in $\mathbb{R}^2$, prove that the Lebesgue measure is equal to triangle is ...
1
vote
2answers
55 views

Additivity for closed intervals

Let $\mathscr{F}$ be the set that contains all finite, disjoint union of closed or open intervals. Given $A\in \mathscr{F}$, let $\mu(A)=1$ if, for some positive $\varepsilon$, $A$ contains the ...
0
votes
1answer
100 views

Inequality on monotone function integration

Let $f$ be a nonnegative monotone function defined on the $[0,1]$. Prove that $$\mu_L(\{w\in I;f(w)>\alpha\})<\dfrac1\alpha\int_0^1fdx$$ where the measure is the Lebesgue measure, and the ...
1
vote
1answer
368 views

Fact about measurable functions defined on $\sigma$-finite measure spaces.

$\bf{\text{Background:}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure space, and $X$ be a Banach space. A function $f:\Omega\to X$ is simple if it assumes only finitely many values. That ...
0
votes
1answer
38 views

Proving that $E_{1} \cup E_{2}$ is measurable if $E_{1}$ and $E_{2}$ are measurable

While proving that $E_{1} \cup E_{2}$ is measurable if $E_{1}$ and $E_{2}$ are measurable, one step I encountered was $m_{e}(T \cap E_{1}) + m_{e}(T \cap E_{2} \cap E_{1}^{c}) \geq m_{e}(T \cap ...
1
vote
1answer
90 views

Riemann Stieltjes integral on $[0,1]$

I am looking for a hint or feedback on what I've already done, not a full solution So say we have the function defined on the unit interval by: $$ \alpha\left(\frac{1}{2}\right) =1, \alpha(t)=0, ...
1
vote
1answer
44 views

Possible measures of a set $E$ when for any interval $I$, $\mu(E\cap I) = \int_I f$

I'm trying to solve the following problem (self-study) regarding Lebesgue measure in $\mathbb{R}$: Let $E \subset [a,b]$, and suppose there is a continuous function $f:[a,b]\to\mathbb{R}$ such that ...
0
votes
1answer
75 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
1
vote
1answer
45 views

$m^*(A) = m^*(A + t)$

Define $m^*(A) = \inf Z_A$ as the outer measure of $A \subseteq \mathbb{R}$ where $$Z_A = \left\{\sum_{n=1}^{\infty}|I_n| : I_n \text{ are intervals}, A \subseteq \bigcup_{n=1}^{\infty}I_n\right\} ...
1
vote
1answer
84 views

Null sets in $\mathbb{R}$

We know $A \subseteq \mathbb{R}$ is null if given $\epsilon > 0$, there exists intervals $\{I_n\}_{n \geq 1}$ such that $$ A \subseteq \bigcup_{n=1}^{\infty} I_n \text{ and } ...
0
votes
1answer
39 views

Ring of sets - Representation of elements

I want to prove: If $\mathcal R$ is a ring of subsets of some non-empty set $X$ and $A_1,\cdots,A_N \in \mathcal R$ then there is some $M \in \mathbb N$ and $B_1,\cdots,B_M$ such that $B_i \cap B_j = ...
2
votes
1answer
67 views

A question about a proof concerning sigma algebras of open sets and rectangles.

While proving $\sigma(\mathcal J_\mathrm{rat}^{n,o})=\sigma(\mathcal O^n)$ in the book "Measures, Integrals, and Martingales" the author does the following: If $U \in \mathcal O^n$ we have $$U= ...
8
votes
0answers
316 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
2
votes
2answers
259 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
1
vote
1answer
357 views

Continuity of the Lebesgue function

If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ ...
2
votes
0answers
181 views

total variation measure vs. total variation of its distribution function

Let's say that $f:[0, \infty)\rightarrow \mathbb{R}$ for simplicity, although the question is also intended for a variety of other types of domains. (Probably interval is all that is required.) ...
0
votes
2answers
473 views

How to find a sequence of step functions that converges to $\frac{1}{\sqrt{x}}$?

Definition:Let $v:(a,b)\to\mathbb{R}$. If there exists a partition $P$ of the interval $(a,b)$ such that $v$ is constant in each subinterval of $P$, we say that $v$ is a step function. Let ...
4
votes
1answer
70 views

Does the empty set need to be explicitly listed as an element of a sigma field?

When enumerating the sigma fields that can be generated by a set $X$, does the empty set need to be explicitly listed as an element of each of the sigma fields or do the facts that $X$ is an element ...
2
votes
1answer
56 views

What is the First order logic representation of a measure space?

How can a measure space be written in first order language? I'm presuming that it must be many sorted. How in particular can the measure be written?
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2answers
85 views

Is this Proof Concerning Measure Theory Valid?

So I know it is true that if $\mu(E)\gt0$ (where $\mu$ is Lebesgue measure of $\mathbb{R}$) then there exists $z,\ x,\ y$ such that $x,y,z\in E $, $x\neq y$ and $\frac{x+y}{2}=z$. I was just curious ...
0
votes
3answers
67 views

Why and when $\lim_{r\to0}\int_{\partial B(x,r)}u(y)\;dS(y)=u(x)$?

Let $U\subset\mathbb{R}^n$ be an open set, $x\in U$ and $u\in C^2(U)$ a harmonic function. I would like know what is the theorem that is used to conclude that $$\lim_{r\to0}\int_{\partial ...
2
votes
1answer
67 views

Integral on $\ell^{\infty}$

I begin with measure and integral theory. I want to give answer on the following statement: Suppose $l^{\infty}$ is the Rieszspace of all bounded functions on $\mathbb{N}$. Define ...
5
votes
1answer
251 views

Haar measure on O(n) or U(n)

Every locally compact group has left-invariants haar measures. In particular, the compact groups O(n) and U(n) have them. I was wondering if there is a realization of such a measure on these groups, ...
1
vote
1answer
363 views

Applying measure zero definition to Cantor sets

I just learned about the concept of measure zero in real analysis, i.e. the definition that a set in $\mathbb{R}^n$ has measure zero if for any $\epsilon$ it can be covered by countably many ...
8
votes
2answers
309 views

Subspaces of $L^p$

So studying Qualifying Exam problems in Analysis I cam across this one: For $1\lt r \lt p \lt s \lt \infty$ where $\mu$ denotes Lebesgue measure, a) Construct a subspace of $L^p([0,1],\mu)$ such ...
4
votes
2answers
128 views

Any finite set is a null-set

How can we prove that a finite set is a null-set? Maybe would it be easier to prove that the outer measure of a finite set is $0$? any ideas on how to tackle this problem? thanks,
7
votes
2answers
1k views

Must the (continuous) image of a null set be null?

Say $E \subset [0,1]$ is a null set. Let $f: [0,1] \rightarrow [0,1] $. Do you think $f(E)$ is a null set or not? Just being curious. (DEF): A set $A$ is null if given any $\epsilon > 0$, there ...
1
vote
1answer
165 views

What almost sure convergence means in the context of strong law of large numbers

According to http://en.wikipedia.org/wiki/Almost_sure_convergence#Almost_sure_convergence, a sequence of random variables $X_n$, which are a function of a shared sample space $Ω$, is said to converge ...
2
votes
1answer
1k views

Irrationals in $[0,1]$ does not have measure zero

Show that the set of irrationals in $[0,1]$ does not have measure zero in $\mathbb{R}$ By definition of measure zero, we must show that there exists $\epsilon>0$ such that the set $A$ of ...
0
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1answer
57 views

Does integrability on whole line imply integrability on subset?

Does the integrability of $f\in L_1(\mathbb{R},\mathfrak{B}_{\mathbb{R}},m_L)$ imply that \begin{equation}\int_A f\,\mathrm{d}m_L\end{equation} exists and is finite for $A\subset \mathbb{R}$, without ...
1
vote
1answer
74 views

Atoms in a countable space

Let $(\Omega, \mathcal{F})$ be a measurable space where $\Omega$ is countable. I am trying to prove that there is some partition $\mathcal{P}$ of $\Omega$ such that the $\sigma$-algebra ...
3
votes
0answers
115 views

Upper semicontinuity of a probability measure

Let $m$ be an atomless probability measure on $\mathbb{R}^m$. Consider $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that for all $v \in \mathbb{R}^m$, $x \mapsto f(x,v)$ is ...
0
votes
0answers
26 views

Controlling measures uniformly

Consider $\mathbb{R}^{2}$ and restrict ourselves to only measures which are $\mu\ll\mathcal{L}$, where $\mathcal{L}$ is Lebesgue measure. Let $\mu^{n}$ be a sequence of probability measures and $B$ is ...
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2answers
148 views

The description of the product sigma-algebra for countable products

This is taken from Folland 's analysis book. I basically got stuck on understanding the proof of proposition 1.3 . I will appreciate if anyone can help to clarify. P/S : Can anybody show me ...
10
votes
2answers
4k views

Example of a continuous function that is not measurable

Let $\mathcal{L}$ denote the $\sigma$-algebra of Lebesgue measurable sets on $\mathbb{R}$. Then, if memory serves, there is an example (and of course, if there is one, there are many) of a continuous ...
3
votes
1answer
269 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
6
votes
1answer
279 views

Example of a set $Y$ that has zero Lebesgue measure and a continuous function $f$ such that $f(Y)$ is not a set of zero Lebesgue measure.

Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a ...
3
votes
1answer
146 views

Generated semiring

In Pap E., Handbook of Measure Theory, Vol.1 (Elsevier), p.30, "For any class $\mathcal{F}$ of subsets of $S$ [$S$ is a non-empty base set] there is a smallest semiring containing $\mathcal{F}$, and ...
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votes
2answers
346 views

A pathological example of a differentiable function whose derivative is not integrable

First I'll make a definition: $$\operatorname{Loc-int}(g):=\left\lbrace x\in[0,1] : \exists \epsilon>0\text{ s.t. }\int_{(x-\epsilon,x+\epsilon)\cap[0,1]}|g|dm<\infty\right\rbrace,$$ where $m$ ...
1
vote
1answer
55 views

Caratheodorys Criterion

Let $A$ be a subset of $R^d$ such that $0 < \mu(A) < \infty$. Given $0 < \alpha < 1$, there exists a cube $Q$ such that $\mu(A ∩ Q) \geq α |Q|$, where $\mu$ denotes the exterior ...
3
votes
1answer
60 views

weak convergence : probability measures, is limit finite?

Assume that there is a sequence of measures $\mu^{n}$ all defined on $\mathbb{R}^{N}$, where the sigma algebra is Borel. Furthermore $\mu^{n}$ is assumed to be tight. I know that Prohorov's theorem ...
3
votes
0answers
64 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
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1answer
177 views

If the weighted Lp norm of a measurable function is finite, is the weighted Lp norm of the antiderivative also finite?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} dx < \infty. $$ Define $g : \mathbb{R} \rightarrow \mathbb{R}$ to be $$ ...
1
vote
1answer
115 views

Why is the Hausdorff measure no Radon measure for all $s < n$?

Let $\mathcal{H}^s$ be the s-dimensional Hausdorff measure on $\mathbb{R}^n$. We showed that for all $s > 0$ this is a Borel regular measure on $\mathbb{R}^n$. Now in our lecture notes, we state ...
4
votes
1answer
417 views

Fubini Tonelli's Theorem, Measure Theory

This is a question from Folland's Book, Suppose $$f(x,y)=(1-xy)^{-a}$$ and $a>0$. Check if the following integrals exist and if they are equal: $$\int_{[0,1]\times[0,1]}f ...
2
votes
1answer
158 views

How to show that every bounded variation function on $[a,b]$ is differentiable a.e?

I'm struggling with this question for over a week now. I know the proposition is true, but haven't managed to prove it yet. any suggestions anyone? ($f$ is BV on $I$ if ...
0
votes
1answer
85 views

Layer-Cake for general functions

The Layer-Cake representation of a non-negative measureable function $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ is given by $$f(x) = \int^{\infty}_{0} \mathbb{I}_{\{y\ \in\ ...
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vote
0answers
121 views

Prove that the set N of normal numbers has negligible complement.

I'm taking a graduate course called Real Variables I, without having taken the prerequisite of Real Analysis II, and having taken only Real Analysis I. Therefore, I'm brand new to measure theory and ...