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

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51 views

Show the linear span of the given collection is dense in the space

Show that space generated by $\{1, \sin x, \sin 2x, \ldots \sin nx,\ldots\}$ is dense in $C([0,\pi])$. I am having trouble to show it algebra and get explicit function that separates end points. ...
2
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1answer
107 views

Show: $M\subset\mathbb{R}^n$ Jordan-measurable, iff $vol^*(\partial A)=0$

Show that a bounded subset $A\subset\mathbb{R}^n$ is Jordan-measurable iff and only if $\partial A$ is a Jordan null set, i.e. $vol^*(\partial A)=0$. Here Show some properties of the ...
2
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1answer
50 views

showing that $f=\chi_A$ for some measurable set $A$!

hi need some hints with this question: If $f^n$ is integrable for each $n$ and $\int f^n dµ = c$ for some constant c then show that $f(x) = \chi_A(x)$ for some measurable set $A ⊂ X$. I know that we ...
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1answer
35 views

if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ continuous, $f_i=X \rightarrow \mathbb{R}$ measurable prove $h(x) = g(f_1(x),…,f_k(x))$ is measurable.

if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ is continuous and $F_i=X \rightarrow \mathbb{R}, i = 1,2,...,k$ is measurable Prove that $h(x) = g(f_1(x),f_2(x),...,f_k(x))$ is measurable. So far I ...
1
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1answer
138 views

Is the completion of a saturated measure saturated?

A measure $\mu$ on $(X,\mathcal{M})$ is saturated if every locally measurable set belongs to $\mathcal{M}$. A set $E\subset X$ is locally measurable if $E\cap A\in \mathcal{M}$ whenever ...
2
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1answer
231 views

Show some properties of the (inner/ outer) Jordan-measure

Let $A\subset\mathbb{R}^N$ be a subset. Show that (1) $vol^*(A)=vol^*(\overline{A})$, (2) $vol_*(A)=vol_*(A^°)$. Here $vol^*(A)$ of a bounded subset $M\subset\mathbb{R}^N$ is defined by ...
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1answer
44 views

if $\mathcal{A}$ is a $\sigma$-algebra, for $y \in \mathcal{A}$ show that $\mathcal{A}_y=\{y \cap A|A\in \mathcal{A}\}$ is also a $\sigma$-algebra.

If $\mathcal{A}$ is a $\sigma$-algebra, for $y \in \mathcal{A}$ show that $$\mathcal{A}_y=\{y \cap A|A\in \mathcal{A}\}$$ is also a $\sigma$-algebra. So far I have shown that $\emptyset \in ...
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2answers
95 views

How do we prove that if $M$ is complete, any sequence $x_n \in M$ satisfying $d(x_n,x_{n+1}) < 2^{-n}$ converges in $M$?

I'd like to show that if $M$ is a metric space $(M,d)$, and is complete, any sequence $x_n \in M$ satisfying $d(x_n,x_{n+1}) < 2^{-n}$ converges in $M. My proof is the following: I think that ...
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1answer
136 views

How can I prove that the open interval $(c,d)$ is not of measure zero? [duplicate]

I am currently trying to prove this theorem. I know how to do the closed interval case, it is a contradiction proof that uses the heine-borel theorem. Does anyone know how I can do the open interval ...
2
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3answers
64 views

Proof of if $g$ is continuous and $g(x) = 0$ almost everywhere on the closed interval $[c,d]$, then $g(x) = 0$?

Suppose that $g$ is some continuous function on $[c,d]$. Now also suppose that $g(x) = 0$ almost everywhere on the closed interval $[c,d]$. We would like to prove that $g(x) = 0$, $\forall x\in ...
3
votes
2answers
404 views

Why does an open interval NOT have measure zero?

I am currently working on a proof that requires me to show that an open ball $B_{\epsilon}(x)$ has nonzero measure. I currently have the following proof in my book: "The closed interval $[a,b]$ is ...
1
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2answers
69 views

limit of integrals of increasing functions

Let $\{f_n\}$ be a non-decreasing sequence of integrable functions such that $\lim_{n \to \infty} f_n \geq 0$. If $S = \{x|\lim_{n \to \infty}f_n > 0\}$ is not a null set, show that $\lim_{n \to ...
2
votes
1answer
94 views

Adjoint of an integral operator

I'm reading through a text about integral operators and I've come across the following theorem: Let $k:\mathbb{R}^2\rightarrow\mathbb{C}$ be a kernel, $T:L^2(\mathbb{R})\rightarrow ...
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0answers
62 views

Series of functions as Lebesgue integral with counting measure

Let $f_n:\mathbb{R}\to\mathbb{R}$ for $n=1,2,...$. Then \begin{equation}\sum_{n=1}^\infty f_n = \int_{\mathbb{N}}f_n\text{ d}\mu \end{equation} where $\mu$ is counting measure on $\mathbb{N}$. I ...
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1answer
94 views

Show that $\Psi(S)$ is a Jordan null set

Let $U,V\subset\mathbb{R}^n$ be open sets, $\Psi\colon U\to V$ a $C^1$ diffeomorphism and $S\subset\overline{S}\subset U$ a Jordan null set. Show that then $\Psi(S)$ is a Jordan null set, too. ...
2
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1answer
54 views

Show that measure has a particular property

This is a general question: If I want to show that my measure has a particular property, how can I do this? For instance, my measure is invariant under orthogonal transformations of my sets, what are ...
6
votes
1answer
111 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
3
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2answers
648 views

Lebesgue Integral, Riemann Integral and Integrals of all sorts

I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ...
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1answer
52 views

Borel sigma algebra in topology and R

This is a very basic question about which I am concerned. What is the difference between the Borel $\sigma$-algebra for a topological space and for $\mathbb R$? or they are same?
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3answers
90 views

If $\mu$ is a signed measure, does $\mu(E)=0$ and $F \subset E$ imply $\mu(F)=0$? [closed]

Let $(X, \mathcal{A})$ be a measurable space and let $\mu$ be a signed measure on $\mathcal{A}$. Let $E, F \in \mathcal{A}$, $F \subset E$. Does $\mu(E)=0$ imply $\mu(F)=0$? I couldn't find any ...
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0answers
31 views

a.e. convergence of dilations of a function in L^p

Let $f\in L^p(R^d)$ ($p<\infty$) and $\delta_h f(x):=h^{d/p}f(hx)$ (the normalization is so that $\|\delta_h f\|_{p} =\|f\|_p$). Consider $(h_n)$ a sequence of positive numbers such that either ...
2
votes
1answer
51 views

Integral result, looking for proof or reference

I am looking for a proof or reference to a text which has the following result: Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. Then if $f$ is a measurable function of some sort then if it ...
1
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2answers
64 views

integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0 $. Show that $f$ must be constant. Well, it's clear ...
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1answer
43 views

How to prove a function is in $L_2(G)$

For $G$ is open bounded set in $\mathbf{R}^n$, $L_2(G)$ is defined as $$L_2(G) = \{f:G\rightarrow \mathbf{R} \text{ measurable}: \int |f|^2<\infty\}$$ Now, I want to prove that $f:\mathbf{R} ...
1
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1answer
34 views

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $/iff$ $f^+$ and $f^-$ are measurable

Prove that if (X, $\mathcal{A}$, $\mu$) is a measure space, $f$ is measureable $\iff$ $f^+$ and $f^-$ are measurable$\mathcal{A}$ Where $f^-(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } ...
1
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1answer
37 views

constructing product measure, is this function well defined?

I am reading about product measure, and my book does this: A and B are measurable sets on a $\sigma$-algebra. Often I see that we have to prove that the function is well defined if we construct a ...
4
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2answers
137 views

Cardinality of an algebra

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets. I am asking (1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak ...
5
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1answer
127 views

Qual problem in Analysis

I am having trouble with the following qual problem. Some help would be awesome. Thanks. Let $f$ be a measurable function on $(0, ∞).$ Let $p > 1/2$ and define $g(x) = (x^p + x^{−p})f(x).$ Show ...
4
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1answer
76 views

Dense subspace of the space of measures on the torus $\mathbb{T}$.

Every measure $\mu$ on the torus $\mathbb{T}$ is the weak-$\ast$ limit of a sequence of absolutely continuous measures on $\mathbb{T}$ with $C^{\infty}$ densities. I'd like to see a proof of this ...
1
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1answer
59 views

Showing set is measurable in a product space w.r.t. the product $\sigma$-algebra

Statement: Let $(X,\mathcal{B}, \mu)$ be a $\sigma$-finite measure space, and $f:X\rightarrow [0,\infty]$ is $\mathcal{B}$-measurable. Together with $(\mathbb{R},\mathcal{B}[\mathbb{R}], m)$, the set ...
1
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0answers
69 views

LimSup of Random Variable

I have a seemingly trivial question. Why does $$\forall a\in\mathbb{R},\mathbb{P}(\limsup X_n>a)>0\Rightarrow \mathbb{P}(\limsup X_n=\infty)=1$$ Clearly, we don't have (at least trivially), ...
5
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1answer
81 views

Prove $\lim_{n \rightarrow \infty}$ $\frac{1}{n}$$\int_{0}^{\space n}xg(x)dx=0$

If $g$ is a Lebesgue integrable function in $E=\lbrack 0,\infty)$, prove that $$\lim_{n \rightarrow \infty}\frac{1}{n}\int_{0}^{\space n}xg(x)dx=0.$$ I want to use the absolute continuity of the ...
2
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0answers
79 views

A detail on Lusin's theorem

Suppose that $B$ is a ball of $\mathbb{R}^{m}$, $(m\geq2)$, and $f(x)$ a measurable function on $B$. According to Lusin's theorem, we can find a closed set $F\subset B$ whose complement has a measure ...
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0answers
69 views

Disjoint sets producing strict outer measure inequality

Can you produce a sequence of pairwise disjoint sets $\{E_i\}$ where $m^{*}(\bigcup E_{i}) < \sum m^{*} (E_{i})$? So I realize these sets cannot all be measurable, and must all have finite outer ...
2
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1answer
83 views

Possible mistake in Safonov's lemma

$\newcommand{\measure}[1]{\lvert#1\rvert}$Let $B_{R_0}$ a ball in $\mathbb{R}^d$, $R_{0}>0, 0<\xi<1$ and $\Gamma \subset B_{R_0}$ be a measurable set such that $\measure\Gamma >0$. ...
0
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1answer
264 views

Properties of Sigma Algebras of Information up to a stopping time

first of all i want to ask whether given any two $\{\mathcal{F}_t\}$-stopping times $\sigma, \tau$ is the following properties true: (i) $\mathcal{F}_{\sigma \wedge \tau} = \mathcal{F}_{\sigma} \cap ...
2
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0answers
59 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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1answer
208 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
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1answer
152 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
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0answers
110 views

L2 norm and the Kullback distance

Let $P$ and $Q$ be two probability measures with densities $p$ and $q$ with respect to the Lebesgue measure on [0,1] such that: $0<a\leq p(x)\leq b$, $0<a\leq q(x)\leq b$ $\forall x\in $[0,1] ...
3
votes
1answer
50 views

function $L_p$ iff $1\leq p<2$

Let $X=<0,1>$, take the borel sigma algebra, and the lebesgue measure. Consider $g(x)=\dfrac{1}{x^{\frac{1}{2}}}$. Show that $g\in L_p$ iff $1\leq p<2$. I have done this: ...
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2answers
72 views

function that doesn't belongs to $L_1$, but belongs to $L_p$ for $1<p\leq\infty$

Working on Bartle's book The Elements of Integration I found this exercise: Take $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, with $\mu$ as countable measure and define $f(n)=\dfrac{1}{n}$, prove that ...
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1answer
109 views

Product measure with a Dirac delta marginal

Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like ...
0
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1answer
119 views

bounded function continuous except for a set of measure zero

Let $f$ be a bounded real function on $\mathbb{R}^n$ and $P$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. If $f$ is continuous on $P^c$, then $f$ is Riemann integrable. Is it true? my ...
1
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1answer
81 views

Identify the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains the set [0, 1]

This is a past exam question which I've tried to do this myself, though I'm unsure of the solution. First of all, by the definition of a sigma algebra, I should include $\emptyset$. Then, ...
1
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1answer
34 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
0
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1answer
52 views

Does $L^p$ convergence imply convergence of integrals?

If $L^p-\lim_{t\rightarrow\infty} f_t = f$ ($p > 1$), is it the case that $\lim_{t\rightarrow\infty}\int f_t^p = \int f^p$?
1
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0answers
86 views

[Measure theory]. Proof of inequality of integrals of simple function

I have a question regarding a proof in my textbook. The theorem is as follows : if {$f_n$} is an increasing sequence of non-negative simple functions and $lim_{n \rightarrow \infty}f_n(x) \geq g(x), ...
2
votes
0answers
33 views

Meaning of a probability on a product defined by a transition probability

If $(\Omega,\mathbb{B},P)$ and $(\Omega',\mathbb{B}')$ are sets with sigma algebras and $P$ is a probability on $(\Omega,\mathbb{B})$. If $\nu(\omega,B')$ is a transition probability from ...
6
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1answer
196 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation ...