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

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Why does $\left\{\left|X-\sum_{i=1}^mX_i\right|\le\delta\right\}$ depend only on $X_i,i\ge m+1$, if all $X_i$ are independent?

Assumptions Let $(X_i)_{i\in\mathbb{N}}$ a sequence of independent real-valued random variables and $$B_{m,n}:=\left\{\left|X^{(i)}-X^{(n)}\right|\le2\delta \text{ for all }i\in[n,m-1]\text{ and ...
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47 views

If $\int_A f d\nu = \int_A f d\nu'$ for some $f>0$ and all measurable $A$, does $\nu$ equal $\nu'$?

Let $\nu,\nu'$ be finite measures (same total mass) on $\mathbb{R}^d$ (with Borel sigma-algebra), and $f: \mathbb{R}^d \rightarrow (0,1)$ be a measurable function. Note: $f$ is fixed (this the the ...
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24 views

$f$ be Lebesgue integrable and $F = m\{f > \alpha\}$, then $F$ is right continuous

The following is a part of a problem 18.2 from from Real Analysis, N. L. Carothers: Let $f: \mathbb{R} \rightarrow [0, \infty]$ be integrable and define $F: [0, \infty) \rightarrow [0,\infty]$ by ...
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97 views

Some Problems in the book ‘A course in Probability Theory’, K.L. Chung

Q1.(page10.6) A point $x$ is said to belong to the support of the d.f.(distribution funtion) $F$ iff for every $\epsilon>0$ we have $F(x+\epsilon)-F(x-\epsilon)>0$. The set of all such $x$ is ...
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55 views

If $f_n \to f $ in measure can we say that $\ |f_n| \to |f|$ in measure

If $f_n \to f $ in measure can we say that $\ |f_n| \to |f|$ in measure ( asumme we are in finite measure spaces ). I think it can not be true because $||f_n|-|f||\le |f_n-f|$
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25 views

$f $ Borel measurable $\iff f^+, f^-$ are both Borel measurable

Let $f= f^+-f^-$. Show that $f$ is Borel measurable if and only if $f^+, f^-$ are both Borel measurable. Attempt: Suppose that $f^+$ and $f^-$ are both Borel measurable. Then their difference ...
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59 views

$(f_n)$ Borel, $f(x)=\lim_{n\to\infty} f_n(x) \implies f$ Borel

Prove that if $(f_n)$ is a sequence of Borel measurable functions and if $f(x)=\lim_{n\to \infty}f_n(x)$ exists in $\mathbb{R}$, then $f$ is Borel measurable. In fact $f$ is Borel measurable even ...
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27 views

Subset $A\subset\mathbb R$ such that for any interval $I$ of length $a$ the set $A\cap I$ has Lebesgue measure $a/2$

Is there a subset $A\subset\mathbb R$ such that for any interval $I$ of length $a$ the set $A\cap I$ has lebesgue measure $a/2$? Can it be constructed explicitly?
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37 views

An application of the Dominated Convergence Theorem

I read the following argument. Consider an integrable function $f$ satisfying that: $$ F(x) - F(a) = \int_a^x f(t) dt, $$ for $a \leq x \leq b$ and some function $F$. Then $$ F(x+h) - F(x) = ...
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29 views

Complex Measures: Function Space

Given a locally compact Hausdorff space $\Omega$ and a Banach space $E$. Denote functions with compact support by: $$\mathcal{C}_0(\Omega,E):=\{F\in\mathcal{C}(\Omega,E):\operatorname{supp}F\subseteq ...
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34 views

Example of an outer measure that I don't understand

So I understand everything in this except verifying (04) that it is sub-additive I don't understand the part of the existence of A covering {${B_{nk}}$} and afterwards. Could someone explain this part ...
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32 views

function that send every lebesque measureable sets to lebesque measurable sets then it send measure zero sets to measure zero.

I want to proof that : If $f: \mathbb R \to \mathbb R $ is a function that send every lebesque measureable sets to lebesque measurable sets then it send measure zero sets to measure zero. I do ...
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46 views

$\sigma-$algebra , inverse function

Is a $\sigma-$algebra a set that contains all the subsets of a set? In my lecture notes there isthe following: $$f(x)=\sin x \\ f^{-1}\left (\left [\frac{1}{2}, 1\right ]\right ...
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17 views

example in measure theory that I am not understanding

This example presented by the book I don't understand how did they get that lim $A_n$ = 0. I understand why the limit exist since we have $(....A4 \subset A3 \subset A2 \subset A1)$ and hence the ...
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48 views

$f(x) = f_0(||x||)$, show that $f$ is measurable iff $f_0$ is measurable

Let $f_0:[0,\infty) \to \mathbb{R}$, and let $f:\mathbb{R}^m \to \mathbb{R}$. say that $f(x) = f_0(||x||)$ Prove: $f$ is measurable if and only if $f_0$ is measurable. I tried to solve it myself and ...
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12 views

Adaptedness of modifications under the usual conditions

Suppose we have a filtration $\mathbb{F} $ which satisfies the usual conditioning so it contains all null sets and is right continuous. Now let $X_t$ with $t\in T$ be a stochastic process adapted to ...
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33 views

A measurable function (with complete measure) is sum of two other functions

Let $(\Omega, \mathcal A, \mu)$ be a measure space and let $\overline{\mu}$ denote the completion of $\mu$. I have to show that if $f \colon \Omega \to \mathbb R$ is $\overline{\mu}$-measurable then ...
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145 views

Is $\sin(1/x)$ Lebesgue integrable on $(0,1]$?

Is the function $f(x)=\sin(1/x)$ Lebesgue integrable on $(0,1]$? I know that, as $f$ is continuous on the set, it is a measurable function. However, I'm stumped on how to go on. A nudge in the right ...
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31 views

question about the dominated convergence theorem

The dominated convergence theorem states that if we have have $(u_j)_{j\in \mathbb{N}}$ which are all integrable with respect to the measure $\mu$ and $|u_j|<w$ for an positive integrable function, ...
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43 views

Consider $\mathbb{R}$ with the lesbesgue measure, show that there does not exist an $f$.

Consider $\mathbb{R}$ with the lesbesgue measure, show that there does not exist an $f$. with $\int_{\mathbb{R}}| f |< \infty$ and $\int_{\mathbb{R}}| f-1 |< \infty$ I have no idea where to ...
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53 views

prove that the lesbesque measure of the Cantor set is 0

the cantor set is constructed by removing the middle part of the previous cantor sets. So $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ $C_2 = [0, \frac{1}{9}] \cup [\frac{2}{9},\frac{1}{3}] ...
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32 views

Set of discontinuities of one function is smaller than that of another

Let's say I know that the set of discontinuities of a function $f$, denoted by $D_f$, has measure zero (although I don't believe that fact matters). I think that it follows that the set of ...
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169 views

Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this, in this case I would like to see the proof. Consider some fixed real numbers $\lambda_1\leq\ldots\leq\lambda_N$ ...
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26 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
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46 views

Calculate $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$

I have to calculate (if it exists) $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$. I think I have to use Lebesgue dominated ...
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67 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
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1answer
43 views

Showing that a collection of intervals (see problem) generates the Borel sigma algebra on $(0,1]$

I would be very appreciative if someone could show me how to do this problem so that I can try to get a better understanding of what a Borel sigma algebra is. Examples are how I learn best so seeing ...
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1answer
36 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
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160 views

Exercise 3.40 from Folland, Real Analysis

Let $F$ denote the Cantor function on $[0, 1]$ (see $§1.5$), and set $F(x)= 0$ for $x<0$ and $F(x)=1$ for $x>1$. Let ${[a_n, b_n]}$ be an enumeration of the closed subintervals of $[0,1]$ with ...
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22 views

Show that $S(f^{-1}(C))=f^{-1}(S(C))$

Show that $S(f^{-1}(C))=f^{-1}(S(C))$, where $f:X \to Y$ is a function and $C$ is a non empty family of subsets of $Y$. And $f^{-1}(C):=\{f^{-1}(c)|c \in C\}$ and $S(C)$ is the $\sigma$-algebra ...
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73 views

Regularity of Dirac measure on Baire sets

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
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40 views

Limit of density points is a density point? [closed]

Let $\lambda(A)>0$ for some $A \subset \mathbb{R}^n$ and let $x_n \in A$ be a sequence of Lebesgue density points of $A$ with $x_n \to x \in int(closure(A))$. Must $x$ be a density point as well?
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49 views

Measure of the reciprocal of a Cantor set

I have recently started studying measure theory and as is usual we started out by calculating the measure of the Cantor set. Now I had this question in my mind as to whether the set generated by ...
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25 views

Question about Haar Measure from Halmos

Halmos (Measure Theory, 1950, p. 256) poses the question: Given a Locally Compact Group $G$, compact subsets of measure zero, $C$ and $D$, is the group product, $P=CD$, which is also compact, also of ...
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34 views

How to show that a integrable function is finite a.e.?

I want to show that an integrable function is finite a.e. Then I have to show that if $$\int fd\mu<\infty \implies \mu(\{x\mid |f(x)|=\infty \})=0$$ for a measure $\mu$. My idea is: $$\{x\mid ...
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48 views

Study the convergence of the sequence of functions $f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$ (convergence in measure, pointwise and in $ L^2(R ^d)$

Study the convergence of the sequence of functions $$f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$$ (convergence in measure, pointwise and in $ L^2(\mathbb{R} ^d)$). Let f be a measurable function such ...
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45 views

Differentiation theorem for Radon measures

I have trouble to understand a detail in the proof of the following Theorem: Theorem: Let $\nu, \mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be outer Radon measures, such that $\nu \ll \mu$. Then ...
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23 views

Need help with Lebesgue measures and continuity

Let $E\subset\Bbb{R}$ be a Lebesgue measurable set such that $\lambda(E)<\infty$ and $a,b\in\Bbb{R}$ such that $a<b$. (a) Prove that $f(x):=\lambda([a,x]\cap E)$ is continuous in $[a,b]$. (b) ...
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61 views

Help explain this “atom” and what this $G$ is.

Following are some slides from a lecture that you can watch here (starting at 20:40). It is explaining the Hierarchical Dirichlet Process. https://www.youtube.com/watch?v=PxgW3lOrj60 In the first ...
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72 views

Conditional Expectation of X

How do I calculate the conditional expectation of $E(X \mid X)$ where $E \vert X \vert<\infty$?
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62 views

Show that the distance $D_c$ between densities is symmetric when the densities are related by a linear transformation

The distance between two density functions $p_0$ and $p_1$ is given by $$D_c(p_0,p_1)=\int_{p_0/p_1>c} (p_0-c p_1)\mathrm{d}\mu$$ where $c>1$ is a real number Question: Show that if ...
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16 views

$\int\left|e^{i\langle s-t,x\rangle}-1\right|^2\mu(dx)=2(1-\Re(\varphi_\mu(s-t)))$ for all finite measures $\mu$ with $\mu(\mathbb{R}^n)=1$

Let $\mathcal{B}(E)$ denote the Borel $\sigma$-algebra on $\mathbb{R}^n$ $\mu :\mathcal{B}(\mathbb{R}^n)\to [0,\infty)$ be a measure with $\mu(\mathbb{R}^n)=1$ $\varphi$ be the characteristic ...
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182 views

Measure on the set of rationals

Consider the rational intervals defined as $$[a,b)_Q= \{r : a \leq r < b; r \in \mathbb{Q} \},a,b\in \mathbb{Q}, a<b. $$ Let $A$ be the class of all sets of rationals that can be produced as ...
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80 views

Definition of $\pi$ and d -systems.

Definition: Let $\Omega$ be a sample space. a) A d-system is a family of subsets containing $\Omega$ and closed under proper difference (if A,B $\in\mathcal D$ and A $\subseteq$ B, then B \ A ...
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47 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
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111 views

Difference of elements from measurable set contains open interval

Let $A\subset\mathbb{R}$ be a measurable set s.t $,m(A)>0$. Prove that the set $$B=\{x-y\mid x,y\in A\}$$contains nonempty open interval around 0. I thought to take an interval in $A$, ...
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1answer
27 views

Determining the orthogonal complement of $\{1 \}^\perp$ in $L^2[0,1]$

Consider the space $L^2[0,1]$ of complex valued square-integrable functions $f : [0,1] \to \mathbb{C}$. Let $\langle f, g \rangle = \int_0^1 f \bar{g}$ denote the standard $L^2$ inner product. For $M ...
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49 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
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2answers
48 views

Using the MCT to evaluate the integral of a series

I'm studying for my Measure Theory final and I've come across a question that I can't seem to find an answer for. For each $n \in \mathbb{N}$ set $E_n:=[n,2n]$ and let $f:\mathbb{R} \to \mathbb{R}$ ...
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1answer
113 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...