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

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26 views

Interchanging limits in stochastic order notation definition

A sequence of real-valued random variables $(X_n)$ is said to be $O_P(b_n)$ where $(b_n)$ is a sequence of positive numbers if $$ \lim_{T\to\infty} \limsup_{n\to\infty}P\{|X_n|>T b_n\}=0$$ Is it ...
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0answers
24 views

Measure Theory vs. Decision Theory - problem classification

I am having trouble classifying my problem, and I am seeking some guidance on book advice. I don't know if I have measure-theory problem and/or a decision-theory problem (or other field). I want to ...
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0answers
29 views

Equivalent term for lebesgue measurable set

I am trying to solve the following problem: Given that $E⊆R$ and for every $n∈Z: m^*(E ⋂ [n,n+1])+m^*(E^c ⋂ [n,n+1])=1$, prove that E is lebesgue measurable (meaning, for every $A⊆R : ...
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0answers
56 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
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1answer
43 views

Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a ...
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1answer
42 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
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1answer
43 views

Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.

Let $A\subset \mathbb{R}$ be of positive Lebesgue measure, i.e. $\mu(A)>0$. Is it then true that $\mu(\mathbb{R}\setminus (A+\mathbb{Q})) = 0$? I am quite sure that if $\mu(A)>0$, then $A-A$ ...
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1answer
37 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
1
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1answer
26 views

Taking limits of both sides of inequality in Royden and Fitzpatrick

A theorem in Royden and Fitzpatrick's "Real Analysis" relies on taking limits of both sides. First the theorem: Let $E$ be a measurable set of finite outer measure. Then for each $\varepsilon > ...
3
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0answers
64 views

What is the difference between $\mathcal B(E\times F)$ and $\mathcal B(E)\bigotimes\mathcal B(F)$

What is the difference between $\mathcal B(E\times F)$ and $\mathcal B(E)\bigotimes\mathcal B(F)$ I know the definition of $\mathcal B(E)\bigotimes\mathcal B(F)=\sigma\{A\times B:A\in\mathcal ...
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1answer
34 views

Showing a set is $\mu$-measurable.

I'm trying to show that if $B\subset \mathbb R$, $A\subset \mathbb R$, $A$ is $\mu$-measurable, and $\mu^*(S(A,B))=0$, then $B$ is $\mu$-measurable. Here are the definitions I have to work with ...
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0answers
18 views

Radon Measure and Radon Integrals

Let X be a compact Hausdorff space and $\mu$ a Borel Measure on $\beta(X)$. Show that there is a constant $c\geq 0$ such that $\mid \int f d \mu\mid \leq c\parallel f \parallel_{max}$ for all $f$ in ...
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1answer
56 views

Integral with respect to a non-standard measure

Let $\mu:P(\mathbb R) \to [0,+\infty]$ be a measure defined by: $$ \mu (\{ \tfrac 1n \})= \tfrac 1n $$ and $\mu(E)=0$ if $E \cap \{ \tfrac 1n \}_{n \in N_0} =\emptyset$ Compute $$\int_{\mathbb ...
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0answers
25 views

real analysis and Radon measure

Let X be a compact Hausdorff space and $\mu$ a Borel Measure on $\beta(X)$. Show that there is a constant $c\geq 0$ such that $\mid \int f d \mu\mid \leq c\parallel f \parallel_{max}$ for all $f \in ...
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1answer
36 views

Real analysis radon measure

Show that a Dirac delta measure on a topological space is a Radon measure. Show that the sum of two Radon measures is also a Radon measure. Please help me.
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2answers
186 views

Borel Sigma Algebra

The question is asking to prove that the family: $\{(−a, a) : a \in \mathbb{R}\}$ does not generate the Borel $\sigma$-algebra. It is known that the family $\{(a,b) : a < b\}$ generates the Borel ...
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1answer
26 views

Borel-Cantelli Lemma “Corollary” in Royden and Fitzpatrick

The Borel-Cantelli Lemma in Royden and Fitzpatrick's "Real Analysis" seems to be a sort of "corollary" of the non-probabilistic ones I see online. It says: "Let $(E_k)_{k=1}^{\infty}$ be a countable ...
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1answer
41 views

subspace of Hilbert space is closed if and only if it is weakly closed

Any hints for this question, thank you! Prove that a subspace of Hilbert space is closed if and only if it is weakly closed.
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1answer
25 views

$\min \{||x-x_0|| : x \in V \} = \max \{|<x_0,y> : y \in V^{\bot} \}$

I need some help, please Let $V$ be a closed subspace of Hilbert space $H$, and let $x_0 \in H$. Show that $\min \{||x-x_0|| : x \in V \} = \max \{<x_0,y> : y \in V^{\bot} \}$ thanx in ...
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1answer
21 views

How to get the inner measure of the Vitali set.

I know that this should be very easy, but why exactly is the inner measure of the Vitali set 0?
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1answer
33 views

Uniform Convergence of sequence of measurable functions

I'm trying to work through problems in Royden Fitzpatrick and I'm stuck on the following problem: Suppose $\{f_n\}$ converges to the real-valued $f$ pointwise on E, where each $f_n$ is measurable on ...
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1answer
63 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
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0answers
35 views

Notation in Lp spaces

I have a question about notation. If we have the space $L_p([a,b])$ with $1\leq p<\infty$ and $f\in L_p$ is it true that $\int_a^b \! |f(x)|^p \, \mathrm{d}x < \infty, \forall x\in[a,b]$. I ...
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1answer
52 views

The measure of a rotated 1-dimensional null set

Let $\nu:\mathcal{B}_{\mathbb{R}}\to [0,\infty]$ be the Borel measure defined by $$ \nu(E) \:= \lambda_2\{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \in E\} $$ Where $\lambda_2$ is the Lebesgue measure on ...
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0answers
22 views

Weak Law of Large Numbers

The Weak Law of Large Numbers is often stated with the iid assumption for the underlying RV's. However, I have seen the independence assumption being diluted to the "uncorrelatedness" assumption ...
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1answer
44 views

Show the outer measure of a union is the sum of the measures without Caratheodony

I am attempting the following question: Let $\mu^*$ denote an exterior measure, $\{A_j\}$ collection of disjoint, $\mu^*-measurable$ sets, show for any E: $\mu^*(E \cap (\cup(A_j)) = \sum ...
2
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1answer
64 views

why calling these 'algebra' and 'ring' too?

In measure theory you have 'algebra's' and 'rings' as subsets of the powerset of the underlying set of the measurable space. If I am well informed then you speak of an algebra if it is closed under ...
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1answer
27 views

Smallest Algebra Containing Singletons

$\Omega:=\mathbb N$. What is the smallest algebra containing all singleton $\{\omega\}$, i.e. $\{1\}, \{2\}$, and so on. Any hint, please?
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0answers
24 views

How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
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0answers
15 views

Are they $\pi$ systems?

I am not sure whether the following two systems are closed under finite intersections. $\{(a,b):-\infty<a<b<\infty\}$: I do not think it is if I consider $(0,1)\cap(1,2)=\emptyset\notin ...
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3answers
48 views

Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
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1answer
15 views

Is this true for any $L^p$ space?

Suppose $f\in L^p$ with $1\leq p<\infty$. Let $E_\alpha=\{x\mid|f(x)|>\alpha\}$. Then $$\lim_{\alpha\to\infty}\int_{E_\alpha}|f|^p d\mu=0$$ Any hints would be appreciated.
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2answers
14 views

Proving equality of sigma-algebras

Let $C_1$ and $C_2$ are two collections of subsets of the set $\Omega$. We want to show that if $C_2$ $\subset$ $\sigma$[$C_1$] and $C_1$ $\subset$ $\sigma$[$C_2$], then ...
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1answer
112 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
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1answer
92 views

open sets have measure zero??

Suppose $A \subseteq \mathbb{R}^n$ is an open set. Can we conclude that $A$ does not have measure zero?? I am trying to find an open set with measure zero, but it seems quite hard to construct one ...
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1answer
25 views

f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
3
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2answers
31 views

lebesgue measurable problem

Let $E$ be a Lebesgue measurable subset of $[0,1]$, and suppose that $m(E)>3/4$. Prove that $(-1/2,1/2) \subseteq E-E$. We use $E-E$ to denote the set $\{x-y:x,y \subseteq E\}$
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0answers
39 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
0
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1answer
23 views

For $(a,b)$, if $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) \cap - E)$ then $E$ in $\mathbb{R}$ is measurable

If $m^* ((a,b)) = m^* ( (a,b) \cap E ) + m^*( (a,b) - E)$ for all open intervals $(a,b)$, then $E$ in $\mathbb{R}$ is measurable. How do I prove this? Totally stuck.
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1answer
28 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists ...
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1answer
18 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists ...
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1answer
49 views

Continuous, strictly increasing function that maps a set of positive (lebesgue) measure onto a set of measure zero?

Is there a continuous, strictly increasing (real-valued) function on the interval $[0,1]$ that maps some set of positive (lebesgue) measure onto a set of measure zero? Should I play with cantor ...
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0answers
21 views

Measure Theory Question 3

Let $E$ be a measurable set with $m(E)<\infty$. Show that there is a descending sequence of open sets $\{G_n\}$ so that $E\subseteq G_n$ for all $n \ \epsilon \ \mathbb N$ and $ \lim_{n\to\infty} ...
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3answers
16 views

A question on limit of weak-* convergence of probability measures

Let $(X,\mu)$ be a measure space. Assume $X$ is compact. It is well-known that the space $\mathcal{P}(X)$ of probability measures on $X$ is compact in weak-* topology. Let's consider a sequence of ...
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0answers
32 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
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0answers
38 views

Weak Compactness for measures

Let $\{\mu_k\}_{k>1}$ be a sequence of radon measures on $\mathbb{R}^n$ satisfying $\sup_k\mu_k(K)<\infty$ for each compact set $K\subset \mathbb{R}^n$. Prove that if ...
2
votes
2answers
41 views

Lebesgue integrability and measurable functions

Let $f$ be a nonnegative function on the reals. What does the (Lebesgue) measurability of $f$ have to do with the (Lebesgue) integrability of $\int f$? I've spent some time studying the definition at ...
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2answers
37 views

Collection of half open intervals is an algebra, why?

I have to show, why the collection of all finite unions of such half open intervals $(a,b]$ is an algebra and not a sigma algebra. I know that $−∞≤a≤b≤∞$, and have: $$ (a,b)=\bigcup_{n=1}^∞ ...
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1answer
53 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous?

Which are the conditions for a Lorentz space $L^{p,q}$ to be order-continuous? ( A Banach function space is order-continuous $\equiv$ Increasing sequences of order-bounded positive functions ...
4
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1answer
47 views

Show that $g\in\mathcal{L}^q(\mu)$.

Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that $$\int |fg|d\mu\leq C\|f\|_p$$ for ...