Questions tagged [measure-theory]

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

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Sigma algebra generated by a set of functions

Let, $\Omega = \mathbb{R}^{[0,1]}$, that is the set of all functions from $[0,1]$ to $\mathbb{R}$, equipped with the sigma field $\mathcal{A}$ which makes each one dimensional projection measurable. ...
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Measure, Integration & Real Analysis Sheldon Axler Section 2B Exercise 16

I apologize for the title in advance, but this is a long problem. So, here it is: Suppose $\mathcal{S}$ is a $\sigma$-algebra on a set $X$ and $A \subset X$. Let $$\mathcal{S}_A = \{E \in \mathcal{S} :...
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Is there exist countable union of half-open $(a,b]$ in Interval $(0,1]$

We know that $\mathbb{X} = (0,1]$,and $\mathcal{M} = \{\emptyset,\text{finite unions of disjoint intervals that are open on the left and closed on the right} \}$ is not a $\sigma$-algebra. However,if ...
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Exercise 5.S. the elements of integration and lebesgue measure

I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint? Suppose the function $x\rightarrow f(x,t)$ is $X$-...
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2answers
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Show $f: [0,1] \rightarrow [0, \infty)$ is Lebesgue measurable.

Let $X = [0,1]$. Let $f: X \rightarrow [0, \infty)$. Define $G(f) = \{ (x,y)\in X \times [0, \infty] : y \leq f(x) \}$. Show that $f$ is $m$-measurable iff $G(f)$ is $m^2$-measurable and that $$m^2(G(...
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Folland Theorem 2.40

In the proof of Theorem 2.40 b), it says the proof follows from Theorem 1.19. But in Theorem 1.19, we have $E \subset \mathbb{R}$, not $\mathbb{R}^n$. Why does the proof follow from that ? Thanks!
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Lebesgue integral of $f : [0,1] \to \mathbb{R}$, where $f$ is defined as a piecewise function.

Today during my measure theory lecture, my professor introduced the following problem to us Suppose that $f: [0,1] \to \mathbb{R}$ is defined by letting $f(x)=0$ on the Cantor set $\mathcal{C}$ and $...
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21 views

Why the following set is stable by difference

$(X,A,u)$ is a measure space $f:(X,A)\to(\mathbb{R},B)$ is an $A$_$B$ measurable & $u$-integrable function , where $\mathbb{R}$ is the set of real numbers and $B$ is Borel set $Z=\{Y\in A \mid \...
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what is the final velocity of the combined mass

A 0.25 arrow with a velocity of 12 m/s west strikes and pierces the center of a movable 6.8 kg target. What is the final velocity of the combined mass?
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$X_n \to X \implies f(X_n) \to f(X),$ for a Borel function $f$

$(X_n)_n$ is a sequence of identically distributed random variables, $f:\mathbb{R} \to \mathbb{R}$ a Borel function. Prove that if $X_n$ converges in probability to $X,$ then $f(X_n)$ converges in ...
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What would be the diameter of this set? [closed]

Let, $t \in [0,1]$ and $x,y \in \mathbb{R}$. Let, $f(t) = tx + (1-t)y$, such that, $f:[0,1] \rightarrow \mathbb{R}$. What would $$\textrm{diam}(f([0,1])),$$ where $\textrm{diam}([a,b]) = \sup\{d(x,y):...
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1answer
14 views

Prove that an interval is NOT in the sigma algebra generated by a collection.

I have been thinking about this for a while and could not find an answer. This is a homework question so any hints are appreciated (HINTS ONLY). Let $\mathcal{M}$ be the sigma-algebra in $\mathbb{R}$ ...
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Convolution of signed measure

I am reading this paper on Likelihood computation of Multinomial distributions. I am stuck in a particular step (on page 5) that involves convolutions of distribution functions which I am unable to ...
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1answer
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on the external Lebesgue measure of a set [closed]

if $E \subset \mathbb{R}$ and $r>0$ we define $rE=${$rx:x \in E$}. If $m^*(E)=l$ what is $m^*(rE)$ ? I believe by observation of simple sets that the answer is r*l but i have no idea how to proceed....
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Why is weak convergence of bounded measures defined for $C_b(\mathbb{R}^d)$?

Definition: For any sequence of measures $(\mu_n)_n \subset M_b(\mathbb{R}^d)$ where $M_b(\mathbb{R}^d)$ is the space of bounded measures we say $(\mu_n)$ converges weakly to $\mu$ if for all $f \in ...
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A problem from an introduction to measure theory

This is Exercise 1.5.7 from Tao's An introduction to measure theory.I've proved (1),but I don't know how to solve (2). Exercise 1.5.7.Let $(X,B,\mu)$ be a measure space,let $f_n:X→C$ be a sequence of ...
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1answer
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If $X_n$ is an $L_1$-bounded martingale, show that $\sum_n(X_n-X_{n-1})^2< \infty$ a.s.

I had asked this question previously in the following post but there were no replies. Recently, I found a two page article(free download) with a possible alternate line of proof to the one suggested ...
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27 views

special case for stopping time

I am a little confused about a spacial case of stopping time: Let X = $\{X_n : n ≥ 0\}$ be any stochastic process and suppose that $\tau$ is any random time that is independent of $X$. Then $\tau$ is ...
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26 views

Show that this family of measures is tight

Let $(\mu_t)_{t\in I}$ and $(\nu_t)_{t\in I}$ be families of finite signed measures on a metric vector space $E$ and $\mu_t\ast\nu_t$ denote the convolution of $\mu_t$ and $\nu_t$. Assume $(\mu_t)_{t\...
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1answer
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Why does the Galton-Watson Process dies out a.s. when the mean is less than 1?

The following image describes a Galton-Watson process. Let $\xi_i^n$, $i,n\geq1$, be i.i.d. nonnegative integer-valued random variables. Define a sequence $Z_n$, $n\geq0$ by $Z_0=1$ and $$Z_{n+1}=\...
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express the following map in terms of the counting measure then deduce that it is a measure [closed]

$u(A) = \text{Card}(A)$ is counting measure in $(\mathbb N,\mathcal{P}(\mathbb N))$, And the map is defined by $$v(A) = \sum \frac{1}{(k+1)^2} \hspace{2cm} (k \in A)$$ For $A$ in $\mathcal{P}(\mathbb ...
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Showing a property holds in arbitrary measure space

A measure space $\left(X,\mathcal{M},\mu\right)$ is said to have Property A if for every $A\in\mathcal{M}$ with $\mu(A)>0$ there is a set $B\subset A$, $B\in\mathcal{M}$ such that $0<\mu(B)<\...
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2answers
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The reason for the superiority of Lebesgue measure / Lebesgue integration

I'm learning about Lebesgue measure and Lebesgue integration. I know that Lebesgue integration is superior to Riemann integration in the sense that we can integrate a much larger class of function ...
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41 views

Using Martingales to find expected values.

Suppose we have i.i.d random variables $X_1,X_2,...$ s.t $P(X_i=1)=\frac{1}{2}=P(X_i=0)$. Let $$\Omega=\inf\{n\geq 5|(X_{n-4},X_{n-3},X_{n-2},X_{n-1},X_{n})=(1,0,1,0,1)\}.$$ I would like to use ...
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1answer
15 views

Countable linear combinations of indicator functions has a canonical form and form a vector space

Generalizing simple functions, let us say we have countable linear combinations of indicator functions over $\mathbb{R}^n$, that is $f(x)=\sum_{i=1}^{\infty}a_i\chi_{E_i}(x)$, where $\{E_i\}$ is a set ...
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Continuity of Measure for Decreasing Sequence

Given a measure space $X$ with positive measure $m$, for a sequence of decreasing sets , i.e. , $A_{i} \supset A_{i+1}$, it might not be true that the $m(\cap A_{i}) = \lim m(A_{i})$. Suppose $X$ is a ...
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Is this a topology? (induced by weak convergence)

Let $\mu,\nu$ probability measures on two compact sets respectively $X,Y\subseteq\mathbb{R}^n$. Let $\Pi(\mu,\nu)$ be the space of measures on $X\times Y$ whose first and second marginals are $\mu,\nu$...
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1answer
23 views

Books with Exercises Measure-theory

Before I start: I know this question does not meet the MSE guidelines and it's going to be closed (I have no problem with that. I will even delete it myself if I have my answers). So, I'm just asking ...
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1answer
35 views

Why is $\frac{Z_n}{\mu^n}$ a martingale? (Galton-Watson Process.)

The following images taken from Durrett Pg 200 explain what a Galton Watson Process is and its corresponding martingale $\frac{Z_n}{\mu^n}$. However, I don't see why $\frac{Z_n}{\mu^n}$ is a ...
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1answer
59 views

$f$ continuous $\Rightarrow$ $f$ $\mu$ measurable?

Let $f: X\to [0,\infty]$ be a continuous function and $\mu$ an outer measure on X. For a continuous function I can split it in a sum of step functions, so $$ f(x) = \sum\limits_{i=1}^\infty s_i \chi_{...
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1answer
23 views

Fatou’s lemma equality

I found a sequence of functions that makes Fatou’s lemma be a strict unequality, but I’m not able to find one that makes it be an equality. Can you help me to find one? Thank you
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a closed subset $F$ of $Q^{C} \cap [0,1]$ has continuous function

CAn we find a closed subset $F$ of $Q^{C} \cap [0,1]$ such that $I_{Q^{C} \cap [0,1]}$ (where the indicator function on the irrationals in $[0,1]$) is continuous on $F$ and $m([0,1]-F) < \epsilon$ ...
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1answer
28 views

How to verify the funtion $\Bbb{P}_{a}$ , defined as $\Bbb{P}_{a}(B) = \int_{B} f d\lambda$ for any Borel set B, is a probability measure

I am trying some exercise on a measure theoretic probability text, and want to make sure if I am doing right. The question is: Let ($\Omega$, $\mathbf{F}$ , $\Bbb{P}$) be a probability space. Assume g ...
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Independence of functions of independent random variables.

Suppose $X_1,..,X_n$ are independent random variables and $X_i'$ is an independent copy of $X_i$, then how does one show that $$E[f(X_1,..,X_n)|X_1,..,X_{i-1},X_{i+1},..,X_n]$$ and $$E[f(X_1,..,X_i',....
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1answer
43 views

How to prove that $\int_{[-\pi,\pi]}\log(\vert 1- \exp(it)\vert)\mathrm{d}\lambda(t)=0$?

Let $r>1$ and $\int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{d}\lambda(t)= 2\pi\log(r)$. We want to prove that : $\lim \limits_{r\to 1} \int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{...
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1answer
36 views

When is a locally Borel set Borel?

Let $X$ be a LCH space. Call $E\subseteq X$ locally Borel if $E\cap K$ is Borel for all compact $K\subseteq X$. Evidently, locally Borel sets form a $\sigma$-algebra and every Borel set is locally ...
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Riemann integrable implies Lebesgue Integrale on a Plane

Let $f$ be a bounded real-valued function on $[a,b]$. If $f$ is Riemann integrable, then $f$ is Lebesgue measurable and $\int_{a}^b f(x) dx = \int_{[a,b]}f dm$. The proof of this involves showing that ...
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Borel/ Lebesgue measurable function on $[0,1]^2$

Let $E \subset [0,1]$. Define $D = \{(x,x):x \in E \}$. Let $f= \chi_D$. a) Are $f_x, f^y$ Borel for all $x,y \in [0,1]$? b) Is it true that $f$ is Borel iff $E$ is Borel? c) If we have both $\mu$ and ...
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1answer
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Outer measure of a null set

Outer measure of a null set is zero. proof 1: By definition of outer measure, $m^*(\varnothing)\geq 0.$ Clearly $\varnothing\subseteq (-\frac{1}{n},\frac{1}{n}),\forall n\in \Bbb{N}.$ Therefore, $\...
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1answer
32 views

Meaning of the expression “$\mu$ a.e.”

In the monotone convergence theorem: Let $g_n\geq 0$ be a sequence of measurable functions, such that $g_n \uparrow g\;\; \mu \text{ a.e.},$ i.e. $g_n(\omega) \leq g_{n+1}(\omega)\;\; \mu \text{ a.e.},...
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1answer
38 views

Efron-Stein Inequality proof clarification.

I am currently going through the proof of the Efron-Stein inequality in this set of notes (http://www.econ.upf.edu/~lugosi/mlss_conc.pdf)(P.g. 219-220). However, I have an issue with the final part of ...
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1answer
30 views

Is $\mathcal{F}=\{\{2,4,6\},\{3,4,5,6\}, \Omega, \emptyset\}$ a $\sigma$-algebra over $\Omega = \{1,2,3,4,5,6\}$?

Hello so I'm trying to understand when a set can be defined as a $\sigma$-algebra (I'm new to this :)). I stumbled upon this question: Is $\mathcal{F}=\{\{2,4,6\},\{3,4,5,6\}, \Omega, \emptyset\}$ a $\...
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Measure of an infinite dimensional set

Consider the collection of all infinite products of numbers $x_n$ with $0< x_n< 1$ for every $n$. Even if it might be tempting to say that any infinite product of this kind diverges to zero, it ...
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69 views

If $(X_n)_{n\in \mathbb{N}}$ is a martingale s.t. $\sup_n E[|X_n|]\leq M < \infty$, then $\sum_{n\geq 2}(X_n-X_{n-1})^2<\infty$ almost surely.

I need to prove the above statement. The hint provided was to consider the stopping time $T_l=\inf\{n\in \mathbb{N}||X_n(w)|\geq l\}$ where $l\in \mathbb{N}$ and then show that $E[\sum_{n=2}^K(X_n-X_{...
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39 views

Intuition for the Area Formula (by Federer)

Question I am trying to get an intuition for the area formula as a "generalized" change of variables formula. If $f:\mathbb{R}^m\to\mathbb{R}^n$ is Lipschitz and $m\leq n$ then $$ \int_A g(...
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63 views

Is there a non-atomic probability measure on $\{0,1\}^{\mathbb N}$?

Consider the set $X=\{0,1\}^{\mathbb N}$ together with the discrete topology, and the associated Borel $\sigma$-algebra $\mathcal F$. Is there a probability measure $\mu$ on $(X,\mathcal F)$ that has ...
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1answer
18 views

BV[a,b]$\cap$C[a,b]$\neq$AC[a,b]

Take an $f\in$BV[0,1]$\cap$C[0,1] e.g. the Cantor function. I take the Lebesgue Stiltigies measure of $f$: $$ \mu_f((a,b])=f(b)-f(a). $$ Now I have a finite positive measure, so I can do the Radon-...
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1answer
19 views

Show that if $m^∗(E) < ∞$ and there exist intervals $I_1, . . . , I_n$ such that $m∗E(∆(∪_{i=1}^{n}I_i))< ∞$, then each of the interval are finite

Show that if $m^∗(E) < ∞$ and there exist intervals $I_1, . . . , I_n$ such that $m^∗(E(∆(∪_{i=1}^{n}I_i)))< ∞$, then each of the interval $I_i$ is finite. I have been asked to prove this and I ...
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1answer
42 views

Does $\lim_{a\to 0}\int_1^\infty f\left(\sqrt{a^2+x^2}\right)dx=\int_1^\infty f\left(x\right)dx$ for smooth, rapidly decaying $f$?

Let $f:[1,\infty)\to\mathbb R$ be infinitely smooth and suppose $f$ and its derivatives decay faster than any power of $x$ as $x\to\infty$. In other words, $f\in\mathscr S([1,\infty)),$ where $\...
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1answer
66 views

A question from Rudin RCA

I'm studying Rudin's RCA chapter 7 and get in trouble with exercise 8. Let $V = (a, b)$ be a bounded segment in $R^1$. Choose segments $W_n \subset V$ in such a way that their union $W$ is dense in $V$...

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