Tagged Questions

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

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0
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0answers
4 views

Is the proof proving a mapping is measurable correct?

I am trying to prove that the mapping $(t,\omega) \mapsto \big(\min(\tau(\omega),t),\omega\big)$ is measurable from $\mathcal{B}([0,t]) \otimes \mathcal{F_t} \to \mathcal{B}([0,t]) \otimes ...
4
votes
0answers
12 views

Is the converse of the excision property for outer measure true?

From here on out let us assume $A \subsetneq B$ and that $B$ is measurable. The excision property states that if $A$ is measurable and of finite measure, then $$m^*(B-A)=m^*(B)-m^*(A)\,\,\,\,(*)$$ I ...
0
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0answers
19 views

On Fatou's Lemma

I saw the statement and proof of Fatou's Lemma (proof without monotone convergence theorem, but with bounded convergence theorem- following Stein-Shakarchi). My question is what the Lemma says, or ...
0
votes
1answer
16 views

Let $F$ be the set that contains all lines with two rationals. Show that $M:=\cup_{G\in F}G\subset\mathbb{R}^n$

Let $n≥2$ and $$G:=\{a+tx\mid t\in\mathbb{R}\}, a\in\mathbb{R}^n, x\in\mathbb{R}^n-\{0\}$$ $$F:=\{G\subset\mathbb{R}^n\mid\exists x,y,\in\mathbb{Q}^n\cup G:x\neq y\}$$Let $F$ be the set that ...
0
votes
2answers
9 views

Measurable functions and $L^+$

If $f,g$ are two complex valued measurable functions, then does it imply that $\displaystyle \frac{|f-g|}{1+|f-g|}\in L^+$? (I am sorry if this is a stupid question)
0
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0answers
13 views

Vitali covering theorem for unbounded sets

Ok, let's use the following definitions: Definition: If $B \subset \mathbb R^m$ is a ball, then we denote by $r(B)$ the radius of $B$ and $B^*$ is the ball with the same center as $B$ but with 5 ...
1
vote
1answer
13 views

About Lp spaces [on hold]

Let $(X,\mathcal{M},\mu)$ a measure finite space. Let $f:X\to \mathbb{R}$ a measurable function. Let $$E_n:=\{x \in X : (n-1)\leq|f(x)|< n\}.$$ I want to prove that $f\in L^p(\mu)$ if and only if ...
0
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0answers
16 views

CDF to Borel measure

If I have a right-continuous nondecreasing function $F:\mathbb{R} \to (0,1)$ that tends to $0$ and $1$ as $x$ tends to $-\infty$ and $\infty$ respectively, does $F$ necessarily induce a Borel measure ...
1
vote
1answer
22 views

An outer measure is countable-additive on the measureable sets

Given an outer messure $\mu^*$, i'm trying to prove that for any set $E$, and a countable number of distinct (pairly) $\mu^*$-measureable sets $\{A_i\}_{i=1}^\infty$: $$\mu^*\left(E \cap ...
0
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0answers
16 views

A proof of Hahn decomposition theorem

I find this proof of Hahn decomposition theorem interesting, but also easy until it is said that $F=B_0\setminus\bigcup_{n=1}^\infty B_n$ is negative by construction. I understand that it means that ...
1
vote
1answer
23 views

A question on Hausdorff measure

This question is part of a homework assignment. Considering the hausdorff measure $\mathcal{H}_{2}$ on $\mathbb{R}^{3}$, I need to compute the measure of the unit cube: $A = \{(x,y,z) \in ...
0
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0answers
9 views

In which points of $\mathbb R$ is the distribution of $\mu = \sum_{n = 1}^\infty 2^{-n} \delta_{\phi(n)}$ not continuous?

I am thinking about the following: Let $\phi : \mathbb N \to \mathbb Q$ be a bijection and define a measure on $(\mathbb R, \mathcal B(\mathbb R))$ by $\mu := \sum_{n = 1}^\infty 2^{-n} ...
0
votes
3answers
34 views

Constructing a Measure on the Rational Numbers

I was wondering if there is any known examples of measures on the set of rational numbers besides Lebesgue measure. In particular, an example of a probability measure on $\mathbb{Q}$ would be nice to ...
0
votes
1answer
18 views

Almost a Frechet distribution but not quite yet

I have function as $$\frac{2}{\alpha}x^{\frac{2}{\alpha}-1}e ^{-x^{\frac{1}{\alpha}}}$$ This kind of reminds me of the Weibull and Frechet distribution but not quite because if it were I should be ...
0
votes
1answer
21 views

For a real-valued random variable it holds: $ E(|X|)<\infty\Leftrightarrow \sum_{n\in\mathbb{N}}P(|X|>n)<\infty$

Let $(\Omega,\mathcal{A},P)$ be a measurable space and $X$ be a real-valued random variable on $\Omega$. I want to show that it holds: $$E(|X|)<\infty\Leftrightarrow ...
0
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1answer
17 views

The existence of a measure of finite energy implies a lower bound on Hausdorff dimension

What is the significance of $\mu(x)=0$ and the use of continuity this proof? I am not quite sure about the general direction in the second paragraph.
0
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1answer
23 views

Show that $\lim \sup A _n = \emptyset$ iff $\lim _{n \to \infty } P [\cup _{m=n } A _n ]=0$

Show that $\lim \sup A _n = \emptyset$ if and only if $\lim _{n \to \infty } P [\cup _{m=n } A _n ]=0$ , where $P $ is a probability measure. secondly is it true that $\lim \sup A _n = \emptyset$ ...
2
votes
0answers
47 views

$\sup\limits_{\phi} \int_{[0,1] } \log \phi = \int_{[0,1]} \log f$

Let's say we have a measurable function $f:[0,1] \rightarrow (0, \infty)$. Approximate $f$ from below by a simple function $\phi$, with $\phi(x) > 0$ for all $x$. Then $$\int f = ...
1
vote
0answers
14 views

The inverse of the integration over a ball with radius $\epsilon$

First of all sorry for the nondescript title, but this seemed like the most suitable one. Now let $d\geq2, D\subset \mathbb{R}$ a domain and $G:D\times D\rightarrow[0,\infty]$ continuous. Define ...
0
votes
1answer
24 views

Probability of intersections of independent events with a twist

I'm trying to solve this problem: Let $(A_n)$ be a sequence of independent events. Show that if $I$, $J$, are (finite/countable) disjoint sets, then $$ {\mathbb P}{\Large[}~\bigcap_{i \in I}A_i ...
0
votes
1answer
30 views

An eample of a $\sigma$-finite measure on the Borels of $\mathbb{R}^2$ that isn't a Lebesgue-Stieltjes a measure?

Are there any examples of a $\sigma$-finite measure on $\mathcal{B}(\mathbb{R^2})$ that is not a Lebesgue-Stieltjes measure? By $\mathcal{B}(\mathbb{R^2})$ I mean the Borel Sets of $\mathbb{R^2}$.
1
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1answer
20 views

Equal in distribution but unequal almost everywhere?

If this question has been asked, I apologize but I could not find it. I was wondering if it was possible construct $X$, $Y$ two iid rv's such that they equal in distribution, i.e. $P_X(B) = ...
1
vote
1answer
29 views

Finding a Counter Example - Limits of integrals of an increasing sequence of Borel measurable functions

I need to find a counter example to the following problem. I'm trying to think of some, but maybe I'm not creative. I'm not sure. Let $h$ and $h_1, h_2, h_3, ...$ be Borel measurable functions such ...
2
votes
0answers
19 views

Consistency of Area Axioms

If we want to calculate area of an triangle or equivalently want to calcuate area under curve y=x (x∈[0,1]), we can use two methods: 1)- Assuming we know the area of the square, we can translate the ...
3
votes
1answer
31 views

About generated $\sigma$-algebras (proof verification).

It would be really helpful if anyone would browse through this and tell me if my solution is ok. Here is the question: Let $C \subset 2^X$ be a collection of subsets. Show that for every $K \in ...
1
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0answers
17 views

Limit of Step Functions

It is well known that a measurable function (on a set of finite measure) is limit almost everywhere of a sequence of step functions. I would like to see an example of a measurable function which can ...
0
votes
1answer
12 views

Is the monotone convergence theorem bidirectional?

Say I have $(f_n)$ with $f_1 \le f_2 \le ...$ and I know that $\lim_n\int f_n<\infty$ exists, does that imply $f_n$ converges a.e.? Most formulations I have seen of the monotone convergence ...
1
vote
2answers
26 views

Why $E=\{(x,\alpha)\mid 0\leq \alpha\leq |f(x)|\}$ is measurable?

If $f$ is measurable, why $E=\{(x,\alpha)\mid 0\leq \alpha\leq |f(x)|\}$ is measurable ? to me is by definition but my teacher asked me to prove it (and so that it's not a definition).
1
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0answers
14 views

Representation Theorem for functionals of Continuous Semimartingales

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable. Does it mean that ...
2
votes
0answers
25 views

Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
1
vote
1answer
14 views

Distribution of random variable $Y$ passed throught distributin function of $X$

If \begin{align*} F(x)=P[X \le x] \end{align*} is continuous in $x$, show that $Y=F(X)$ is measurable and that $Y$ has uniform distribution \begin{align*} P[Y \le y]=y, \, 0 \le y \le 1. ...
0
votes
1answer
24 views

On Measurability of random variable

I am studying measurability of r.v. and I am trying to answer the following question. Suppose $X: \Omega \mapsto \mathbb{R}$ has a countable range $\mathcal{R}$. Show $X \in ...
1
vote
2answers
18 views

Show inclusion of $L^p$ spaces in a space of finite measure

Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a space of finite measure we have that $L^{p_2} \subset L^{p_1}$. Could you give me some hints what I could do??
0
votes
1answer
27 views

Measurable Function and Cauchy-Schwarz Inequality

I want to prove using Cauchy-Schwarz Inequality that for any positive function $f$, that $\int_E f \text{ }dm \int_E \frac {1}{f} dm \geq m(E)^2$. Here's what I've got so far. First, note that by ...
0
votes
1answer
15 views

Finite measure on positive integers

Disclaimer: I am sure that this idea is not at all new, but I have had trouble locating content directly related. I humbly accept that this question may be the result of a brain fart. Suppose that ...
0
votes
2answers
20 views

A measurable piecewise function

I want to show that the following functions is measurable: $f:\mathbb{R}\rightarrow \mathbb{R}, f(x) = \begin{cases} \frac{1}{\sqrt {(1-x^2)}} & ,\text{if } x \in [-1,1] \\ 0 & ...
1
vote
1answer
38 views

How is Fubini's theorem used in the following proof?

I'm having trouble to understand exactly how we are using Fubini's theorem in the following proof involving the distribution function, since it newer explicitly involves an integral with product ...
0
votes
1answer
17 views

If $f$ measurable and $f=g$ a.e implies $g$ measurable, then $\mu$ is a complete measure.

It is easy to show that if $\mu$ is a complete measure, then $f$ measurable and $f=g$ a.e implies that $g$ is measurable. However, is it true that if this implication holds, that is $f$ measurable ...
0
votes
0answers
11 views

How is the maximal function of a complex borel measure epual to its total variation measure?

I'm having trouble to understand what the maximal function of the complex Borel measure $\mu $ is. This is from Rudin: Let $B(x,r) $ denote the ball in $\mathbb R ^k $ centered at $x $ and radius $r ...
2
votes
0answers
21 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
1
vote
1answer
27 views

Outer measure is not finitely additive

I know similar questions have been asked before, but I'm looking for clarification of a proof. In Royden's book on real analysis, he proves that every set of positive measure contains a non-measurable ...
1
vote
1answer
37 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
1
vote
0answers
16 views

Borel Measures: Discrete Decomposition

Context The notion of atoms and point masses agree to certain extent. (See Summary on Atoms.) Measures decompose w.r.t. atoms. (See Paper on Atoms.) Here, the goal is a direct approach to decompose ...
1
vote
1answer
35 views

Is a measure, which is equivalent to a discrete measure, also discrete?

Let $(\Omega,\mathcal F)$ be a measurable space. Define a probability measure by $\mathbb P=\sum_{k=1}^\infty\alpha_k\delta_{\omega_k},$ where $(\omega_k)_{k\in\mathbb N}\subseteq \Omega,$ ...
0
votes
0answers
19 views

Integrable equivalence in $\sigma$-finite measure space (Proposition from Bogachev)

I'm reading Bogachev's measure theory book und I don't understand a proof from a proposition in this book. Maybe someone could help me... Here is the statement of Proposition 2.6.2 (ii): Let $(X, ...
1
vote
1answer
7 views

Does the pushforward operator (on measures) preserve surjectiveness?

Let $I = [0,1]$ be the unit interval. Let $\pi: I \to I$ be a Borel-measurable surjective map. Is the pushforward operator $\pi_*: \mathcal P(I) \to \mathcal P(I)$ surjective as well, where $\mathcal ...
1
vote
2answers
33 views

Closed sets with empty interior measure zero

Is the Lebesgue measure of a closed set with empty interior in $\mathbb{R}^{n}$ always zero? Trying to understand something in the math notes that I don't understand, and if the above is true, it ...
0
votes
1answer
14 views

How do I see formaly that $\liminf_{n \rightarrow \infty} |f_n|^p = |f|^p$ for $p \in [1, \infty)$?

Let $(f_n)_{n \in \mathbb N}$ be a sequence of functions s.t $\lim_{n \rightarrow \infty} f_n = f$, where $f_n : X \rightarrow \mathbb R$. How do I see formaly that $\liminf_{n \rightarrow \infty} ...
0
votes
1answer
38 views

Boundedness of sequence of functions

Consider a sequence of continuous integrable functions $\{f_n(t)\}_n$ such that $\ast\ \displaystyle\lim_{n\rightarrow\infty}f_n(t)=0$ for all $t>0$ $\ast\ \{f_n(t)\}_n$ is such that ...
0
votes
0answers
9 views

Find the derivative of $F(t) = \int_{0}^{\infty} \cos(t^2 h(x))e^{-x} \lambda(dx) \ \ (t \in \mathbb R)$ using derivative theorem.

Let $(X, \mathcal E, \mu)$ be a measure-space and $I \subseteq \mathbb R$ an open interval. Let $f: X \times I \rightarrow \mathbb R$ be a function. Consider $$F(t) = \int_X f^t(x) \mu(dx)=\int_X ...