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

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How to give a good upper bound on tail probability for $P\{|\frac{R_n}{\sqrt{n}}-1| \ge \varepsilon\}$?

Suppose $X_1,X_2,\ldots$ is a sequence of i.i.d. standard normal random variables. $R_n=\sqrt{X_1^2+\ldots+X_n^2)}$. How could I prove $P\{|\frac{R_n}{\sqrt{n}}-1| \ge ...
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72 views

Relying two points by an almost-geodesic omitting a singular set a.e.

I failed to give an appropriate title to the question, so any suggestion for a better title is welcome: Here's the question: I would like to prove the following result: Given $\varepsilon>0$, a ...
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49 views

A set with positive measure has a point for which every open interval around it has positive measure

Let $\mu$ be a measure on $(\mathbb{R},\mathcal{B})$. Let $B \in \mathcal{B}$ be such that $B \in (-\infty,\infty) $ and $\mu(B) >0$. Give a proof or else give a counter example to the assertion: ...
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109 views

Show that it is Lipschitz

Let $E \subset \mathbb{R}^d$ be Lebesgue measurable und let $\phi (t)=m \left ( \Pi_{i=1}^{d} (-\infty , t_i ) \cap E \right )$. I have to show that $\phi $ is Lipschitz. Could you give me some ...
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31 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
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27 views

Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT. However, I couldn't find another example when the measure is finite. Could ...
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45 views

The measure of the boundary being zero implies the set is measurable.

Assuming our set, $E\subset\mathbb{R}^2$ such that $m(\partial E)=0$ (where $m$ is Lebesgue measure), why does this imply that $E$ is Lebesgue measurable?
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34 views

Does it stand that $\lim \inf |f_n|^p=|f|^p$ for this reason?

We have that $f_n, f \in L^p, 1 \leq p < +\infty$, $f_n \rightarrow f $ almost everywhere and $||f_n||_p \rightarrow ||f||_p$ . Do we have that $\lim \inf |f_n|^p=|f|^p$ because of the following?? ...
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52 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
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29 views

Is the average of a dense orbit ergodic for shift function?

Let $\sigma$ be the shift function in the space of two-sided infinite sequences of $\{0,1\}$, $X=\{0,1\}^\mathbb{Z}$ equipped with product topology. We know that there is some point $x\in X$ with ...
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34 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
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2answers
54 views

Approximate $f$ by simple functions

Consider the measure space $([0,1)^2, \mathcal{B}([0,1)^2, \lambda)$ where $\lambda$ is the Lebesgue measure on $[0,1)^2$. Put $f:[0,1)^2 \to \mathbb{R}, \,\,\, f(x,y) = x + 2y$. I would like to ...
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28 views

New measure constructed out of supremum of old measure

This is a homework exercise. Let $(X,\mathcal{A}, \mu_1)$ be a measure space. Define for $A\in\mathcal{A}$ $$ \mu_2(A) = \sup \;\{\mu_1(B):B\in\mathcal{A},B\subset A, \mu_1(B) < \infty \}. $$ Show ...
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19 views

Correctly defined measure

I need to show that the measure is unique (correct definition). Prove that the function $\lambda: \sigma(\mathcal{A}\cap V) \rightarrow [0,1]$ such that $\lambda[(A\cap V)\cup(B\cap V^c)]:=\mu(A)$ ...
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48 views

Banach Spaces: Pointwise Limit vs. Uniform Limit

Agreement All notions are up to null sets. Limits are meant by simple functions. Problem Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider bounded measurable ...
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69 views

Product measures and absolute continuity

For $j=1,2$, let $\mu_j,\nu_j$ be $\sigma-$finite measures on $(X_j, M_j)$ such that $\nu_j<<\mu_j$. Prove that $\nu_1 \times \nu_2 << \mu_1 \times \mu_2$. Here, does it suffices to prove ...
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60 views

Caratheodory: Inner vs. Outer

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
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34 views

Is every Lebesgue measurable function bounded on a set of positive measure

Let $f$ be a Lebesgue measurable function from $[0,1]\to\mathbb{R}$. Let $\mu$ be Lebesgue measure. Does there exist a measurable set $B$ with $\mu(B)>0$ and an $M>0$ such that for all $x\in B$, ...
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32 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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30 views

For each $\epsilon >0$ there is a $\delta >0$ such that if $\mu(E)<\delta$ then $\int_E |f_n|d\mu<\epsilon$ for all $n$.

Suppose $(f_n)$ is a Cauchy sequence in $L^1(X,\Sigma,\mu)$ where $(X,\Sigma,\mu)$ is a measure space. Prove that for each $\epsilon >0$ there is a $\delta >0$ such that if $\mu(E)<\delta$ ...
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55 views

Nonmeasurable Functions

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded & Non-Step & Non-Measurable ...
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35 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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58 views

Computing moments

given $\int_{-\infty}^{+\infty} \! e^{-tx^2} \, \mathrm{d}\lambda x = \sqrt{\pi/ t} $ I have been asked to compute the moments $\int_{-\infty}^{+\infty} \! x^{2n} e^{-x^2} \, \mathrm{d}\lambda x $ ...
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32 views

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B \in \Sigma _1 , B _2 \in \Sigma _2 \} $

Show that the product $\sigma $-algebra is generated by the $\pi $-system $\{B _1 \times B _2 : B_1 \in \Sigma _1 , B _2 \in \Sigma _2 \} $ Let $(X , \Sigma _1) $ and $(Y , \Sigma _2 ) $ be two ...
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37 views

If $X:=(X_1,X_2)$ has density $e^{-(x_1+x_2)}1_{\mathbb{R}_{\ge 0}^2}(x_1,x_2)$, then $X_1+X_2$ and $X_1/X_2$ are independent

Let $X_1$ and $X_2$ be real-valued random variables, such that $X:=(X_1,X_2)$ has the density $$f_X(x_1,x_2)=\begin{cases}e^{-(x_1+x_2)}&\text{, if }x_1,x_2\ge 0\\0&\text{, ...
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101 views

Show that the Fourier Transform is differentiable [duplicate]

This might be a silly question. For $f$ an integrable, complex-valued function, its Fourier transform is $$ \hat{f}(s) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-isx}f(x)\, \mathrm{d}x $$ I ...
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1answer
31 views

Almost sure convergence clarification

Let $\Omega=(0,1]$, $\mathbb{P}$ be Lebesgue measure, $n\in\mathbb{N}$ be $n = i + 2^j$, where $j = \lfloor\log_2(n)\rfloor$ and $0 \leq i \lt 2^j$. Let's say I have a random variable ...
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1answer
25 views

Two notions of absolute continuity

If ν is a signed measure and µ a positive measure, we say that ν is absolutely continuous w.r.t. µ if µ(E) = 0 ⇒ ν(E) = 0. If |ν| is a finite measure then this ...
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33 views

Find area of unit square using outer Hausdorff-measure

We put $$\eta_{\delta}(E) = \inf\left\{ \sum_{i \in \mathbb{N}} \text{diam }U_i : E \subset \bigcup_{i\in \mathbb{N}} U_i \text{, and diam }U_i\in(0,\delta] \right\}$$ and for $E\subset \mathbb{R}^2$ ...
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50 views

Existence of distinct points with rational difference in lebesgue measurable set

Let $X \subset [0,1]$ be Lebesgue measurable with $\mu(X)>0$. Show that there exist two (distinct) points $a, b \in X$ with $a-b \in \mathbb{Q}$. I've thought about this for a while but can't seem ...
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33 views

Help understanding the equivalence of these two statements

Help understanding the equivalence of these two statements Let $\Omega $ and $S $ be sets and $Y : \Omega \mapsto S $ $\Sigma $ is a $\sigma $-algebra on $S $ $X: \Omega \mapsto \mathbb R $ Now I ...
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36 views

Using Zorn's lemma to show a set is measurable.

If $S_j\subset \mathbb{R}^n$ and for all $j\in \mathbb{N}$ the set $S_j$ is measurable, show that $\bigcup S_j$ and $\bigcap S_j$ are measurable. I think I can use Zorn's lemma to show that this is ...
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31 views

Proving that sum of two measurable functions is measurable for conditional expectation

I'm trying to show something that seems pretty simple: $\mathbb{E}[aX + Y | \mathcal{G}] = a\mathbb{E}[X | \mathcal{G}] + \mathbb{E}[Y | \mathcal{G}]$ where the conditional expectation is defined ...
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16 views

Show the inclusions $\mathcal L^r(\lambda) \subseteq \mathcal L^s(\lambda)$ and $L^s(\lambda) \subseteq \mathcal L^r(\lambda)$ doesn't hold.

Suppose $0 < r < s < \infty$ and $\lambda$ is the Lebesque measure on $(\mathbb R, \mathcal B(\mathbb R))$. I want to show that $\mathcal L^r(\lambda) \subseteq \mathcal L^s(\lambda)$ and ...
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23 views

Showing that a measure is lower continuous.

Here is the problem and my attempt. Let $\mu$ be a measure defined on sets $E_1, E_2, \ldots$ such that $E_{n+1} \subset E_n$ for all $n$. Additionally, let $\mu(E_1) \lt \infty$. Show that ...
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2answers
84 views

Lebesgue Integral: Convexity

Given a finite measure $\mu(\Omega)<\infty$. Consider a complex function $f\in\mathcal{L}(\rho)$. From the Riemann integral it is evident that: $$\frac{1}{\mu(\Omega)}\int_\Omega ...
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42 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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47 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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23 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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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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29 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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56 views

What are boundary effects in measure-theory?

I often read the term "boundary effects" which seem to be the reason to look at the interior $A^°$ and closure $\bar{A}$ of Borel subsets $A$ separately. What is so special about the boundary and what ...
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34 views

Coupling of r.v.

I am trying to answer this question. If $X$ and $Y$ are random variables on $(\Omega, \mathcal{B})$, show \begin{align*} \sup_{A \in \mathcal{B}} |P[X \in A] -P[Y \in A]| \le P[X \neq Y]. ...
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54 views

Is the supremum of two-variable measurable function always measurable

Problem : [Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be two measurable spaces and let $f\geq 0$ be measurable with respect to $\mathcal{A} \times \mathcal{B}$. Let $g(x)=\sup_{y\in Y} f(x, y)$ and ...
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24 views

Consider two singular measures $ m$ and $v$ and $v$ is absolutely continuous with respect to $m$ show that $v=0$

Consider two singular measures $m$ and $v$ on a measurable space $(X,\mathcal{A})$ and $v$ is absolutely continuous with respect to $m$, i.e., $(v<<m)$. Show that $v=0$.
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24 views

Show that, for all $\delta < \mu(S) $, $\delta >0 $, exists a subset $T$ of $S$ such that $\mu(T) = \delta$.

Let $ \mu $ the lebesgue measure on $\mathbb{R}$ and let $S$ a subset of $\mathbb{R}$ with $\mu(S) > 0 $. $(a)$ Show that, for all $\delta < \mu(S) $, $\delta >0 $, exists a subset $T$ of ...
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25 views

Set of points that makes a series converge absolutely is measurable.

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ a sequence of measurable functions. Prove that the set of points $x\in\mathbb{R}$ making the series $\sum_n f_n(x)$ converge absolutely is a measurable set. ...
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51 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if ...
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48 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
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48 views

A measurable piecewise function

I want to show that the following functions is measurable: $f:\mathbb{R}\rightarrow \mathbb{R}, f(x) = \begin{cases} \frac{1}{\sqrt {(1-x^2)}} & ,\text{if } x \in [-1,1] \\ 0 & ...