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

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Change of variable (or measure)?

Hi Everyone: I am reading a book and there is a kind of "change of variable" they make that I do not understand fully. This is what they do: let $B(x,r)$ be a ball of $\mathbb{R}^{N}$ $(N>1)$, ...
2
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1answer
49 views

Show that $\frac{1}{x^4 \sin^2 (x) +1} \in L^1([0, \infty))$

This is question 10.20c from Apostol's Mathematical Analysis. Basically, I am trying to show that $$ f(x)=\frac{1}{x^4 \sin^2(x)+1} \in L^1([0,\infty)) $$ I know that for some value $k$, I can ...
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1answer
88 views

“Hidden” axiom of choice?

Let $\mu$ be a measure on $S$ such that: $\mu\left(\emptyset\right)=0$ and $\mu(S)=1$ if $X\subseteq Y$, then $\mu(X)\leq\mu(Y)$ $\mu\left(\{a\}\right)=0$ for all $a\in S$ if $X_n$, ...
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0answers
27 views

Lebesgue-Stieltjes measure

Is the following reasonment correct? There is a sort of duality between non-decreasing functions and Borel outer measures. In particular, given a non-decreasing function ...
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1answer
29 views

Set function is a measure

Show that the set function $\mu$, defined on subsets $A \subset \mathbb{N}$ defined by $$ \mu(A) = \sum_{n \in A} 2^{-n}$$ is a measure. The sum of the empty set is defined to be zero and $\mu ...
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2answers
41 views

Measurable Subsets + Caratheodory Measurability

1.) What can go wrong if one assigns a measure to more subsets, especially to all subsets? (I would like to understand the subtleties behind) I imagine the first problem is to give the new subset ...
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1answer
39 views

A well-defined operation on measure algebra

Let $(X,\cal{M},\mu)$ be a measure space, and for $E,F\in \cal{M}$ write $E \sim F$ iff $\mu(E \Delta F)=0$. Let $\widetilde{\cal{M}}$ be the set of equivalence classes in $\cal{M}$ for $\sim$; for ...
2
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1answer
32 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
2
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1answer
35 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
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1answer
23 views

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
22 views

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
14 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
52 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 ...
4
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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
22 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
20 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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0answers
19 views

Bounded linear operator is weak-* continuous

can anyone help me I need to show that : Any dual operator $T^{**}$ is weak-* continuous if $T\in\mathcal{L}(X,X)$ where $X$ is a reflexive Banach space. thanx!
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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 ...
0
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1answer
24 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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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
63 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 $ ...
4
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3answers
58 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
15 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
31 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 ...
0
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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 ...
1
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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
22 views

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
42 views

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
39 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
36 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
17 views

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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0answers
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
25 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 $\mu$ ...