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

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0
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1answer
15 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$
1
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1answer
22 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
3
votes
1answer
23 views

Convergence for every measurable set

Let $(f_n)$ non-negative measurable functions such that $f_n\to f$ and $\int f_n\to \int f<\infty$. We have to prove that $\int_E f_n\to \int_Ef$ for each $E$ measurable. I know that if $f_n\to f$ ...
0
votes
1answer
37 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
0
votes
2answers
15 views

Predual of $l^1(\Gamma)$

Let $\Gamma$ be an uncountable index set. For example $\Gamma=\mathbb R$. Let $l^1(\Gamma)$ be the set of functions with countable support and finite sum: $$ \sum_{a\in\Gamma}|f(a)|<\infty. $$ The ...
2
votes
0answers
7 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 $$ ...
3
votes
0answers
29 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 ...
2
votes
1answer
27 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 ...
1
vote
1answer
37 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 ...
0
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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 ...
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 ...
2
votes
1answer
22 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 ...
0
votes
1answer
46 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
votes
0answers
17 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 ...
1
vote
1answer
17 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 ...
0
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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 ...
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 ...
0
votes
1answer
52 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 ...
3
votes
1answer
48 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 $\left\{x_n\right\}_1^\infty$ ...
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votes
2answers
54 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 ...
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
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
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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) ...
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 ...
8
votes
0answers
62 views
+50

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. ...
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 ...
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
28 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
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1answer
32 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 ...
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
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
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 ...
2
votes
2answers
19 views

A measure theory problem

Consider Lebesgue measure $\lambda$ restricted to the class $\mathscr B$ of Borel sets in $(0,1]$. For a fixed permutation $n_1, n_2, \dots$ of the positive integers, if $x$ has dyadic expansion ...
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 ...
1
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0answers
26 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
1
vote
1answer
38 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
3
votes
1answer
33 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
0answers
16 views

On extension of real valued functions from $(0,1)^{\infty}$ onto $[0,1]^{\infty}$

Let $\lambda$ be the standard Haar measure in $[0,1]^{\infty}$. Let $f$ be a bounded real-valued function on $(0,1)^{\infty}$ such that the set of all discontinuity points of $f$ has ...
2
votes
1answer
35 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
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
38 views

Which are the measurable sets with respect to this construction of an outer dirac-type measure?

Let $x \in \Omega$ and $\mathcal{A} \subset\mathcal{P}(X)$ an at most countable set with $\emptyset, \Omega \in \mathcal{A}$. Let $$ \delta_x: \mathcal{A} \to [0,\infty], \quad A \mapsto ...
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 ...
2
votes
2answers
20 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
0
votes
1answer
17 views

Probability, approximation by simple functions, boundedness and non-negativity.

In my probability class various results were announced requiring that a certain random variable was bounded. The specific example that interests me is the following: if $Y$ is $\mathcal{G}$-measurable ...
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
31 views

Transformation shift measurable

How to prove that this transformation is measurable? $\sigma:B(n)\rightarrow{B(n)} $ $\sigma(x)(k)=x(k+1)$ $\sigma(...,x_{-1},x_{0},x_{1},...)=(...,x_{0},x_{1},x_{2},...)$ where $B(n)$ with ...
1
vote
0answers
19 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
1
vote
1answer
21 views

Fatou's Lemma conditions for strict inequality

Under what conditions do we have equality (resp. strict inequality) in Fatou's Lemma? If the sequence $f_n$ is convergent, then it is obvious that equality holds. Is it the only case? There are some ...
-2
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) $A_1, A_2, ..., \limsup A_{f(n)} \subseteq \limsup A_n.$ ...