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

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Hausdorff measures and densities

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals" Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with ...
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1answer
43 views

What is this theorem about measurable functions saying?

Theorem: Let $(\Omega,\mathcal{F})$ be a measurable space and let $f:\Omega \rightarrow Y$ be a given function. Let $\mathcal{A}$ be a collection of subsets of $Y$. If $f^{-1}(A) \in \mathcal{F}$ ...
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1answer
36 views

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu<\infty$ exists. In ...
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45 views

Sequence that converges as for the norm but not almost everywhere

How can I find a sequence that converges as for the norm but doesn't converge almost everywhere, in some space $L^p$ ?? Could you give me some hints ??
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57 views

Does L1 and nonnegative imply bounded almost everywhere?

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a nonnegative function, such that $f\in L^1(\mathbb{R})$. Does this imply that $f$ is bounded almost everywhere?
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30 views

Convergence in a Measurable set

Let $E$ the set of all $x\in[0,2\pi]$ at which $\{\sin (nx)\}$ converges. This implies that $E$ is measurable? Thanks you all.
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29 views

Finding countable compact set s.t $\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$

Im trying to find a countable compact set such that $$\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$$ I tried thinking about Koch curve, sierpinskii gasket and carpet, Bedford-McMullen carpet and ...
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26 views

Smith-Volterra-Cantor set remove “m”th interval

Is it possible to determine the measure of the Cantor set by removing the middle "m"th interval (m=1,2,3,4,...) from [0,1]? For example, removing middle 3rd from [0,1] gives measure 0; removing ...
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45 views

Union of sets as the union of disjoint sets - Does the proof $\forall n\in \mathbb{N}$ implies the proof for infinity?

I managed to prove that: $$\displaystyle\bigcup_{i=1}^n A_i=A_1\cup(A_1^c\cap A_2)\cup(A_1^c\cap A_2^c\cap A_3)\cup\dots\cup(A_1^c\cap\dots\cap A_{n-1}^c\cap A_n)$$ for $\forall n \in\mathbb{N}$. ...
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84 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
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59 views

Assumptions involving product spaces

Suppose a random variable $X$ is distributed in $\mathbb{R}^{n}$ and we are given that $X' = (X_{1}', X_{2}')$ for $X_{i}$ distributed on $\mathbb{R}^{n_{i}}$. In general, what assumptions can I make ...
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49 views

$M\subset\mathbb{R}^n$ measurable. Show: There is a null set $N \subset \mathbb{R}^n$ and compact seq $K_m$ with $M=N\cup\bigcup_{m\in\mathbb{N}}K_m$.

Assignment: Let $M\subset\mathbb{R}^n$ be lebesgue-measurable. Show that, there is a null set $N \subset \mathbb{R}^n$ and a sequence $(K_m)_{m\in\mathbb{N}}$ of compact subsets $K_m \subset ...
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43 views

If $B\subset\mathbb{R}^n$ is measurable and $(x_l)_{l\in\mathbb{N}}$ is a bounded family, so $(B + x_l)$ is pairwise disjoint, then $\mu(B)=0$.

Assignment: Show that: If $B\subset\mathbb{R}^n$ is Lebesgue-measurable and if there is a bounded family $(x_l)_{l\in\mathbb{N}} \subset \mathbb{R}^n$ so that the family $(B + ...
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1answer
54 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for ...
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1answer
47 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
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1answer
64 views

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in ...
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76 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
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1answer
50 views

every $\sigma$ algebra is a monotone class

I couldn't understand the monotone class theorem because of this lemma: "Every $\sigma$ algebra is a monotone class." How i can prove it?
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32 views

given $E_1, E_2, E_3, …$ prove that the measure of {$x \in X :$ x belongs to infinite number of sets $E_k$} is $0$

Say I have a $\sigma$-algebra $\mathcal{A}$ over a set $X$ and a measure $\mu$. Let $E_1, E_2, E_3, .... \in \mathcal{A}$ such that $\sum_{k=1}^\infty \mu(E_k)$ < $\infty$. let B = {$x \in X ...
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27 views

If $f(\cdot,y)$ is measurable and $f(x,\cdot)$ is continuous, $\{x:|f(x,y)-f(x,0)| \leq \epsilon, \; \;\forall y <\delta\}$ is measurable

Suppose $\mu(X) < \infty$ and $f : X \times [0,1] \rightarrow \mathbb{C} $ is a function such that $f(\cdot,y)$ is measurable for each $y \in [0,1]$ and $f(x,\cdot)$ is continuous for each $x ...
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62 views

Example of algebra that is not a $\sigma$-algebra

I understand that an algebra $F \subset 2^\Omega$ is called a $\sigma$-algebra if it additionaly satisfies: $(A_i)_{i \in \mathbb{N}}$ with $A_i \in F$ pairwise disjoint, then also $\cup_{i \in ...
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34 views

Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

! (http://i.imgur.com/Zwt1m1n.png) I need to do the question at the top of this image. I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim ...
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1answer
32 views

Where does the following intuition about $G_\delta$ sets fail?

Where does the following reasoning that $\mathbb{Q}$ is supposedly a $G_\delta$ set fail? "Proof": $\mathbb Q$ may be covered by selecting open sets $O_n$ such that $m(O_n)<\frac{1}{n}$ for ...
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35 views

Basic measure theory: Composite functions and points of nondifferentiability

I have a function $V(x(t))$. $x(t)$ is continuous, but not everywhere differentiable w.r.t. $t$. What can we say about $V$ at these points of non-differentiability? To explain I have included some ...
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44 views

A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne ...
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3answers
359 views

convolution of characteristic functions

Suppose $A$ and $B$ are measurable subsets of $\mathbb{R}$ of finite positive measure. Show that the convolution $\chi_A*\chi_B$ is continuous and not identically $0$. Use this to prove that $A+B$ ...
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“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ...
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55 views

Exercise on measure theory, (verification and suggestion)

Hi everyone I'd like to know if the following is correct and also I'd appreciate any suggestion to improve the argument. Thanks in advance For every positive integer $n$, let $f_n:{\bf{R}}\to ...
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1answer
33 views

Limit of integral over set measurable

If $A\subset[0,2\pi]$ is measurable, prove that $$\lim_{n\to\infty}\int_A \cos (nx)\ dx=\lim_{n\to\infty}\int_A \sin(nx) \ dx=0$$ Please, any suggestions are welcome.
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61 views

Approximating simple summable function in measure space with countable base

Let $f:X\to \mathbb{Q}+i\mathbb{Q}\subset\mathbb{C}$, $f\in L_1(X,\mu)$ be a Lebesgue-summable function taking only finitely many values $y_1,\ldots,y_n\in \mathbb{Q}+i\mathbb{Q}$ on the sets ...
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1answer
75 views

Product of Absolutely Continuous Measures is Absolutely Continuous

I am stuck on this problem from Folland's Real Analysis, Second Edition: For $j = 1, 2$, let $\mu_j, \nu_j$ be $\sigma$-finite measures on $(X_j, \mathcal{M}_j)$ such that $\nu_j <\!\!< \mu_j$. ...
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56 views

Showing Convergence

Let $(X,M,\mu)$ be a measurable space and $f$ be a real valued integrable function on $X$. Let $E_n=\{x\in X: f(x)\geq nq\}$ for every $n\in \mathbb{N}$ and fixed $q>0$ . Show that ...
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26 views

(Hints please) Constructing a measurable set with the following property.

If $\delta >0$, $I_\delta=(-\delta,\delta)\in\mathbb{R}$, and $0\leq\alpha\leq\beta\leq1$, what hints do you have that would help me figure out how to construct a measurable set ...
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23 views

Showing Convergence in measure with some condition. [closed]

Let $(X,M,m)$ be a finite measurable space and $\{f_n\}$ be a sequence of real valued measurable functions on $X$ . Let $$E_n=\{x\in X : f_n(x)\ne 0\}$$ for every $n\in \mathbb{N}$ . Show that if ...
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50 views

Integral Measures: Variation

Given a measure $\lambda\geq0$. Regard a real function $h:\Omega\to\mathbb{R}$ with $h\in\mathcal{L}$. Consider the real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation ...
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34 views

Showing Convergence in $L^p$ norms

Let $X$ be a finite measure space and $1\le p<\infty$ and $\{f_n\}$ be a sequence in $L^p(X)$ such that coverge to $f$ in $L^p(X)$ . If there exists constant $K$ such that for every $n\in ...
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35 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
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23 views

Moment generating function of bounded variables

According to the answer of this question a moment generating function exists if the random variable $X$ is bounded. The proof is not quite obvious to me. More formally, let $(\Omega,\mathcal{A},\mu)$ ...
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32 views

Analogue of Weierstrass Approsimation Theorem

The following theorems are well knows: Weierstrass Approximation Theorem Given a continuous function on $f\colon [a,b]\rightarrow \mathbb{R}$, there exists a sequence of real polynomials, which ...
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What is a function in $f\in L^n(\mathbb{R}^n)$ but $g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$ is not in $L^\infty$

What is a function in $f\in L^n(\mathbb{R}^n)$ but $$g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$$ is not in $L^\infty$. I have no idea where to start. Apparantly this is related to the ...
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94 views

Borel-Cantelli (proof and application)

Hi I was reading the second volume of the Tao's Analysis book and in one exercise he's asking for a proof of Borel-Cantelli If we have a sequence $s_n\in \Omega$ of measurable sets s.t. ...
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112 views

Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but improper Riemann integral exists

Show that the Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but the improper Riemann integral $\int_{0}^\infty \frac{\sin x}{x} dx = \lim_{t\to\infty} \int_{0}^t \frac{\sin ...
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1answer
61 views

Folland Exercise 3.3: Stuck + Possibly missing $\sigma$-finite hypothesis?

This is Exercise 3.3 from Folland's Real Analysis, Second Edition stated exactly as it appears in the text: ''Let $\nu$ be a signed measure $(X, \mathcal{M})$. $(a)$ $L^1(\nu) = L^1(|\nu|)$. $(b)$ ...
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(Hints please) Prove that $\psi(E)=(\mu\times\lambda)(E)$ for every $E\in S\times T$.

Given that $(X,S,\mu)$ and $(Y,T,\lambda)$ are $\sigma$-finite measure spaces with the measure $\psi$ defined on $S\times T$ such that $\psi(A\times B)=\mu(A)\lambda(B)$ whenever $A\in S$ and $B\in ...
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31 views

How to calculate expectation of Cantor distribution without using p = 0.5

The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following: X is characterized by ...
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1answer
43 views

Non-differentiable in a null set

This is a problem from Stein's real analysis book that I have been working on. Show that exists a non-negative integrable f in $\mathbb{R}^{d}$ so that $\liminf_{m\left(B\right)\rightarrow0,x\in ...
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1answer
38 views

Is it sufficient to check only open intervals in order to prove that a real function is measurable?

Let $f : \mathbb R \to \mathbb R$. We say that $f$ is measurable if, for every $S \in \mathcal B$ where $\mathcal B$ is the Borel algebra on $\mathbb R$, we have that $f^{-1}[S] \in \mathcal B$. I ...
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1answer
32 views

Proove a fact about $\sigma$-algebras in preparation for Kolmogorov's zero–one law

Let $(J_n)_{n\in\mathbb{N}}$ be finite sets which monotonically increase to $I\cong\mathbb{N}$ and $(\mathcal{A}_n)_{n\in\mathbb{N}}$ be a family of $\sigma$-algebras. I want to show that it holds ...
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1answer
40 views

Convergence of a sequence of functions integrated over a sequence of measures

I have real-valued functions $\{f_n\},f$ on a subset $X\subset \mathbb R^n$ that are equicontinuous and I have Borel measures $\{\mu_n\},\mu$. I have that For each fixed $m$, $\int f_m d\mu_n\to\int ...
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1answer
40 views

What do we mean when we say that a function $f$ takes the value $ \infty $?

What do we mean when we say that a function $f$ takes the value $ \infty $? In measure theory it is common to let mappings take values in the extended real number system. But still it doesn't make ...