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

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continuous integral function

Let $K$ be a countinuous and bounded on $\mathbb{R}^n$ and let $f$ be Lebesque integrable on. a) show that $$g(t) = \int_{\mathbb{R}^n} K(tx)f(x)dx$$ is conituous and well defined. b) suppose that ...
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1answer
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convergence in $L^1$ and convergence $\mu$ a.e imply product convergence in $L^1$

Another old exam problem in measure theory im not sure about. Let $(X,A,\mu)$ be a measure space and $f,g, f_n, g_n$ measurable functions on $X$ such that: $(f_n)$ converges to $f$ in $L^1(\mu)$ and ...
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1answer
16 views

Sum of integrals converge or not!

I have an old exam problems I'm trying to solve $$ \sum_{k = 1}^\infty \int_{-R}^0 \frac{x^k}{k!}dx$$ When $R <\infty$ it seems like dominated konvergence and then change the order of the ...
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0answers
15 views

A basic question on integration by parts in general measure space

Suppose $f_1$ and $f_2$ are measurable functions in a general measure space with measure $\mu$. Is there any standard way to calculate $$\int_{A} f_1 f_2 d\mu$$ where $A$ lies in the sigma algebra
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2answers
53 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
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1answer
67 views

Nowhere dense set…

Let $A_n$ be a subset of continuous functions on $[0,1]$ given by: $A_n$ = {$f∈C[0,1]$:there exists $x∈[0,1]$ such that $|f(x)−f(y)|≤n|x−y|$ for all $y∈[0,1]$}. Show $A_n$ is nowhere dense, and use ...
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1answer
23 views

Everywhere continuous extension of a almost everywhere continuous function

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure. If $f$ is continuous outside a set $N$ of $\mu$-measure 0, does there exist an everywhere continuous $g$ such that $f = g$ on $X ...
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0answers
21 views

one Sufficient condition for absolute continuity

Suppose that $f$ is continuous on $[a,b]$, $f'(x)$ exists for every $x \in (a,b),$ and $f'(x)$ integrable. Prove that $f$ is absolutely continuous. How to proceed ?
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2answers
62 views

A problem on calculating integral

Show that the integral $$\int_{0}^{1} \frac{1}{x} \left|\cos \frac{1}{x^2}\right|\ dx$$ is finite. I plotted the graph, but it looks like it is infinity.
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1answer
47 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
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1answer
20 views

Proof that the class of measurable functions is closed under taking limits.

I am working through some revision notes, and I have come across this proof in my lecture notes that the class of $\mathcal{F}$-measurable functions is closed under limit operations: Let $\{f_n\}$ ...
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1answer
30 views

References for a second course in probability theory

I need a probability book that treats all the arguments from the point of view of the measure theory and the Lebesgue integral. I've the basis of "naive" probability theory and of measure theory so I ...
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1answer
26 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
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0answers
17 views

Equivalent definitions of Lebesgue Measurability (Rudin and Royden)

I'm reading Royden's real analysis 4th edition, and he defines a real set $E$ to be lebesgue measurable if, for all real sets $A$, $m(A)=m(A∩E)+m(A∩E^c)$. Here, $m$ is the outer measure of a set. I ...
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1answer
66 views

measure space problem.

Let $(\Omega,\mathcal{F},\mu) $ be a probability space. Let $\delta>0$ and for each $n\in \mathbb{N}$. Let $A_n \in \mathcal{F}$ satisfy $\mu(A_n)\ge\delta$. Prove that the set $A_\infty $ ...
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3answers
59 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
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0answers
17 views

Leibniz's Derivative Rule for Integral in Measure Theory

I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce ...
3
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1answer
73 views

A Baire category question

Let ${f_n}$ be a sequence of real valued continuous functions converging pointwise on $\Bbb R$. Show that there exists a number $M>0$ and an interval $I \subset \Bbb R$ such that $\sup\{ |f_n(x)|:x ...
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0answers
11 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
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0answers
21 views

A basic question on lebesgue integration

I want to prove that the function $2x\sin(\frac{1}{x^2}) - \frac{2}{x}\cos(\frac{1}{x^2})$ is not integrable in [-1,1]. I know how to prove $\frac{1}{x}$ is not integrable (lebesgue) in $[0,1]$. How ...
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1answer
25 views

For measurable $f$ and continuous $g$ on $\mathbb{R}$, $f \circ g$ is measurable?

Here measurability is in the context of Lebesgue measure. So if we suppose $f$ defined on $\mathbb{R}$ is measurable and $g$ on $\mathbb{R}$ continuous then $f \circ g$ is necessarily measurable? I ...
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0answers
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Measure Theory and Topology [duplicate]

Let $X$ be an uncountable set with the discrete topology. 1) Let $X^*$ be the one point compactification of $X$. what is $C(X^*)$ (i.e set of continuous functions on $X^*$). 2) What are the Borel ...
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0answers
30 views

A basic problem on bounded variation

If $a > 0$ let $$f(x) =\left\{\begin{array}{ll} x^{a} \sin (x^{-a})&\text{if } 0 < x \leq 1\\ 0&\text {if }x=0 \end{array}\right.$$ Is it true that for each $0 < \alpha < 1$ ...
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0answers
32 views

Measure inequality implies integral inequality?

Let $f$ and $g$ be non-negative, integrable functions on a measure space with measure $\mu$, and suppose there is some constant $c > 0$ such that for every $t \geq 0$, the inequality $\mu(\{f \geq ...
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1answer
29 views

Measure Theory - Convergence of functions with bounded integrals

A question I came across. Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Let $f_1,f_2,\dotsc:X\to\mathbb R$ be measurable functions such that $n^2\cdot\lVert ...
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1answer
20 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...
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0answers
14 views

(L1* ∩ L2*) = (L1 ∩ L2)* for all languages L1 and L2 over the alpabet Σ={A,B} Is it true or false and why?

plz answer me Determine whether each of the following statements is true or false. If a statement is false, give a counterexample..... 1- $(L_{1}^{*} \cap L_{2}^{*}) = (L_{1} \cap L_{2})^{*}$ for ...
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0answers
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Measure Theory - Lebesgue Integral over non- $\sigma$-finite spaces

In most courses on Measure Theory the Lebesgue Integral is introduced initially for simple functions on finite spaces, then for general functions on finite spaces and finally for general functions on ...
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2answers
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Outer measure and Caratheodory's criterion

Suppose $m^*$ is an outer measure in Caratheodory's sense on the space $X$, which satisfies $m^*(\emptyset)=0$, $A\subseteq B\implies m^*(A)\le m^*(B)$, and $m^*(\bigcup_n A_n)\le\sum m^*(A_n)$. We ...
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0answers
40 views

Measure Theory - An identity for the Lebesgue Intgral

I'm trying to solve the following exercise in Measure Theory: Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Prove that for every $0\leq f\in L^{1}(\mu)$ it holds that: ...
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1answer
19 views

On construction of a Hamel basis which is also a Bernstein set

After having received the answer, I did some googling work and found a proof of the existence of a Hamel basis which is also a Bernstein set on Nonmeasurable Sets and Functions, page 39, Theorem 4. ...
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1answer
37 views

Lebesgue Measure - positive measure sets not containing intervals

I've encountered two statements regarding the Lebesgue measure that don't exactly contradict each other, but seem to me to be a little bit unintuitive when regarded with respect to one another. The ...
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1answer
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Determining the measure (zero) from the measure (zero) of the intersections with translations

Problem $E\subseteq\mathbb R$ is a (not-necessarily measurable) set. $a+E=\{a+x\colon x\in E\}$. If the Lebesgue-measure $m(E\cap(a+E))=0$ for all $a\in\mathbb R\setminus\{0\}$, is it true that ...
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1answer
33 views

Why is the measure of this set 0?

Williams has the following note in his book Probability with Martingales: Lemmma 5.2b simply states that I don't see why $\mu(\{L\neq U\})=0$. I tried doing a proof by contradiction (If ...
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8 views

Minkowski Content-2

Here is a link to the previous question about minkowski content: Minkowski Content My new question is as follows: do $n$-dimensional manifolds have $n$ dimensional Minkowski Content? For example: ...
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0answers
22 views

If $\nu$ is a complex measure, then $L^1(\nu) = L^1(|\nu|)$

I am trying to prove the following statement from Folland: If $\nu$ is a complex measure, then $L^1(\nu) = L^1(|\nu|)$ and if $f \in L^1(\nu)$, then $\left| \int f \; d \nu \right| \leq \int |f| \; d ...
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1answer
27 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
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1answer
21 views

Projection of measurable sets

If $ X $ and $ Y $ are metric spaces, $ f: X \rightarrow Y $ is lipschitzian and $ H^k $ is the Haussdorf measure, it is easy to check that $ f(A) $ is $H^k $-measurable whenever $ A $ is $H^k ...
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0answers
14 views

Show that the graph of a convex function is above any tangent plane

In proving jensen inequality one use that the graph of a convex function is above any tangent plane. I've been reading Property of convex functions and Tangent line of a convex function. But what ...
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0answers
29 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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1answer
39 views

Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
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1answer
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Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
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27 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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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 ...
3
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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 ...
4
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1answer
44 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 ...