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

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2
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0answers
5 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
2
votes
1answer
18 views

Total variation distance is complete

For a given measurable space $X$, $\mathcal{P}(X)$ denotes the space of all the probability measures on $X$. The total variation distance $\rho$ on $\mathcal{P}(X)$ is defined by: for $\mu, \nu \in ...
1
vote
0answers
29 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
3
votes
0answers
29 views

Characterizing a union of intervals.

Find the measure of the set of real numbers in $(0,1)$ whose binary expansions contains zeroes in the odd positions, such that $x = 0.k_1k_2k_3\ldots$ Checking odd positions one at a time: If $k_1 ...
0
votes
0answers
52 views

What is the extent of the streaks covering a square? [closed]

Let $N$ be a unit square, $1 < A <\sqrt{2}$ real number, we put a strip with width $A$ to the square randomly. I would like to determine the measure of the strips, that cover the square. I don't ...
3
votes
0answers
28 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
1
vote
3answers
129 views

Give a example in Lebesgue integral

Please help me to solve the following problem that is in the Lebesgue integral discussion Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that ...
2
votes
1answer
26 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
6
votes
1answer
186 views

For $E \subseteq [0, 1]$, $m(E) > 0$, show that there are $\alpha$ and $\beta$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

This was originally a proof verification question, but I have since moved the proof to an answer as discussed on meta. I still welcome comments on the proof as well as any alternative proofs. ...
0
votes
0answers
26 views

Continuous convergence [on hold]

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
0
votes
1answer
30 views

Can every measure be normalized in order to be a probability measure?

Let $\mu$ be a measure on $(X,\mathcal{A})$. Is it possible to normalize $\mu$ in order to get a probability measure? My idea is to set $$ \mu'(A):=\mu(A)/\mu(X)~\forall~A\in\mathcal{A}. $$
1
vote
0answers
9 views

Proof of Kolmogorov zero-one law in measure-theoretic setting

I have met, in some paper, the following form of the Kolmogorov zero-one law used: If $A\subseteq 2^\Bbb N$ is a subset of Cantor space such that when $x,y\in 2^\Bbb N$ are such that $x,y$ differ ...
0
votes
1answer
39 views

Question about Measure Theory [on hold]

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
0
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0answers
14 views

measure equality on two sigma algebras also holds on the combined sigma algebra?

we have the following setup: $Q, P \text{ are measures on the }\sigma-\text{algebras } \mathcal{F} \text{ and } \mathcal{G} $. Let $P(A)=Q(A) \forall A\in\mathcal{F}$ and $P(B)=Q(B) \forall ...
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votes
2answers
53 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
-1
votes
1answer
43 views

proving a statement based on probability theory [on hold]

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
0
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1answer
50 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
1
vote
0answers
50 views

proving a statement on Measure theory [on hold]

Consider $(\Omega, U, \mu)$ be a measure space and X be an integrable function and for $A$, $\{A_n\}\in \mathscr{U};n\in \Bbb N$ I need to show that $\int_{A_n}X d\mu \to_{n\to \infty}\int_A Xd\mu$ ...
2
votes
1answer
88 views
+50

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
0
votes
2answers
61 views

Radon-Nikodym derivative of Measures [closed]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia, lecture notes with this blog post by John Baez mentioning it as it ...
0
votes
1answer
37 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
16 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
36
votes
4answers
4k views

The sum of an uncountable number of positive numbers

Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many ...
1
vote
1answer
24 views

Countable unions of Vitali sets…

Let $A \subset \mathbb{R}$ be sets of positive Lebesgue measure. Let $\Gamma$ be a countable dense subgroup of the additive group $\mathbb{R}$. Consider the partition of $\mathbb{R}$ canonically ...
3
votes
1answer
46 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
2
votes
1answer
30 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
0
votes
0answers
13 views

convex function with Hessian measure $D^2 f \leqslant \lambda$ $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
0
votes
0answers
28 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\emptyset \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K ...
1
vote
1answer
21 views

Continuity of the function defined by it's measure

The question is as follows: Let $A\subset \mathbb{R}$ be a measurable set with $0<m(A)<\infty$. Define $f$ on $\mathbb{R}$ by $f(x)=m(A\cap(-\infty,x])$. Show that this is continuous. My ...
2
votes
0answers
35 views

Convergence of the integral of a product of functions.

Let $\phi:\mathbb{R^n}\to\mathbb{R}$ be a Lebesgue-measurable function, with the property that for every $n$-dimensional cube $Q$ in $\mathbb{R^n}$, we have $$ \left|\int_{Q}\phi(x)dx ...
2
votes
1answer
28 views

Let $\mathcal R$ be a $\sigma$-ring, then: $\{E\subset X; E\cap F \in \mathcal R\text{ for every } F \in \mathcal R\}$ is a $\sigma$-algebra.

I'm trying to Solve the following question: Let $X$ be a non empty set and $\mathcal R$ be a $\sigma$-ring from subset of $X$. Prove that: $$S=\{E\subset X; E\cap F \in \mathcal R\text{ for ...
0
votes
0answers
10 views

Which of the convolution of integrable functions is continuous?

Let $G$ be a locally compact group with Haar measure $μ$, and let $S$ and $T⊂ G$ be two measurable subsets with $0 < μ(S) < ∞,0 < μ(T) < ∞$. Show that the map $ x → μ(S ∩ xT)$ from $G$ to ...
0
votes
0answers
13 views

Radon measure is a Haar measure

Let $G$ be a locally compact group, $H$ a dense subgroup, and $μ$ a Radon measure on $G$ such that $μ(hA) = μ(A)$ holds for every measurable set $A ⊂ G$ and every $h ∈ H$. Show that $μ$ is a (left) ...
4
votes
0answers
49 views

Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.

It is well known that a sphere minimizes the surface area to volume ratio since it reaches equality in the Isoperimetric Inequality. I'm trying to prove that no other closed surface has this property. ...
2
votes
1answer
36 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
1
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1answer
19 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
3
votes
1answer
341 views

Besicovitch Covering Lemma

We just finished our unit on covering lemma's in my analysis class and my professor proved both the Vitali and Besicovitch covering lemma's (for finite and infinite coverings) using balls. He ...
0
votes
1answer
23 views

Measure Theory (Defining Measureability)

just a very basic measure theory question from the book by Bartle. Let (a) be the statement "For every $\alpha\in\mathbb{R}$, the set $A_\alpha=\{x\in X: f(x)>\alpha\}$ belongs to X." (X is a ...
4
votes
1answer
26 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
1
vote
1answer
31 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
1
vote
2answers
34 views

Basic measure theory question about $\sigma$-algebra

Let $Y, Z$ be random variables and $G$ be a $\sigma$-algebra. Page 69 of Shreve's Stochastic Calculus for Finance II says "because both $Y$ & $Z$ are $G$-measurable, their difference $Y-Z$ is as ...
3
votes
2answers
232 views

Dominated Convergence Theorem for Sets

This was an interesting question, which gives the analog of the better known Dominated Convergence Theorem for Lebesgue integrable functions. Suppose $E_{n}\to E$ pointwise (e.g. the indicator ...
1
vote
1answer
15 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
4
votes
2answers
68 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
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votes
1answer
62 views

Proposition on limsup

Suppose $\exists$ function $f: \mathbb{N} \to \mathbb{N}$ s.t. as $n \to \infty$, $f(n) \to \infty$. Prove that $\forall$ events (or sets) $A_1, A_2, ..., \limsup A_{f(n)} \subseteq \limsup A_n.$ ...
1
vote
1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
1
vote
1answer
54 views

Probability of tail event using Kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
2
votes
1answer
40 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
1
vote
1answer
132 views

What kind of f(n)'s make the limsup statement is true? What kind don't?

What kind of $f(n): \mathbb{N} \to \mathbb{N}$'s make the ff statement true? What kind don't? $\limsup A_{f(n)} \subseteq \limsup A_n$ where $n \in \mathbb{N}$ (*) Well obviously the answers to ...
1
vote
1answer
296 views

Analysis - Fourier Transforms - show that convolution of characteristic functions is continuous

I would appreciate any instruction on the following exercise from real and complex analysis: Suppose $A$ and $B$ are measurable subsets of $\Re^1$, having finite positive measure. Show that the ...