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

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Explicit construction of Haar mesure on the p-adic number field

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. Let $\mathcal B$ be the smallest $\sigma$-algebra containing all the open subsets of $\mathbb{Q}_p$. Can we prove ...
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2answers
51 views

If $u, u'' \in L^2(0,1)$, is it true that $u' \in L^2(0,1)$?

Let $u \in L^2(0,1)$. If $$u'' \in L^2(0,1)$$ is it true that $$u' \in L^2(0,1)?$$ Why yes/not? If $u, u'' \in L^2(0, 1)$ do not imply that $u' \in L^2(0,1)$, how can I show that $u' \in L^2(0,1)$? ...
5
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2answers
38 views

If $f:[a,b]\to\mathbb{R}$ is increasing, does it maps Borel sets to measurable sets?

Suppose $f:[a,b]\to\mathbb{R}$ is strictly increasing and left-continuous. Does it follow that $f$ maps Borel subsets of $[a,b]$ to Lebesgue measurable subsets of $\mathbb{R}$? My intuition tells me ...
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1answer
29 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
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21 views

questions about Folland real analysis chapter 1 exercise

Here, E is a Lesbegue-measurable set on the real line. This is the exercise 30, 31 of p. 40 of Folland real analysis. I solved these problems when E is of finite measure, but the problem requires ...
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0answers
14 views

Solving a problem using of Chebyshev's Inequality

Let $f \in L_{1}(\mu)$ and let $M \gt 0$ such that $$|\frac{1}{\mu(E)}\int_{E}f d\mu| \le M$$ for every $E \in S$ with $0 \lt \mu(E) \lt \infty$. Show that $|f(x)| \lt M$ for a.e $x(\mu)$. Let ...
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1answer
28 views

Finding a new Probability measure

Let $\Omega = \{-2,-1,1,2\}$ and $\mathbb P :\mathscr P(\Omega)\to[0,1]$such that: $$ \mathbb P (F)=\frac{\# F}{\# \Omega}=\frac{\# F}{4} $$ and let $X$ be a random variable such that ...
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0answers
23 views

Measure-theoretic conditional expectation

While working on a homework problem, I am baffled by the following statement: Let $(X,\mathcal{M},\mu)$ be a finite measure space, $\mathcal{N}$ a sub-$\sigma$-algebra of $\mathcal{M}$, and $\nu = ...
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1answer
15 views

meaning of $f_{\chi_{E}}$

given $(X,\mathcal{M})$ a measurable space, I Have $E \subset X$ and $\chi_{E}$ is an indicator function. then what is meant by $f_{\chi_{E}}$ ? I am not very clear with this notation and meaning.
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22 views

For positive measures $\nu_j$, is $\left(\sum_1^\infty \nu_j\right)(E) \le \sum_1^\infty \nu_j(E)$?

I am trying to solve the following homework problem, where the notation $\nu \perp \mu$ means that $\nu$ and $\mu$ are mutually singular: Suppose $\{\nu_j\}$ is a sequence of positive measures. If ...
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1answer
37 views

How can I solve like this exercise in measure theory [on hold]

If $J=\{[a,b[$ : $a \le b$ : $a,b \in R\}$ and $F$ is an continous increasing bounded function on $R$ , and if we put $λ([a,b[)=F(b)-F(a)$ prove that : $$λ(\emptyset) = 0 $$ and if the union of ...
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1answer
47 views

Can someone clarify something in Fubini's theorem please?

In my notes I have a version of Fubini's theorem which differs from the other forms of it I've seen which seem to all be like the one found on wiki here. Here is the version I have in my notes; Let ...
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1answer
20 views

borel measurable functions and measurable functions

Say you are given the Lebesgue measure on the real line and a Lebesgue measurable function $f$. Here Lebesgue measure is a complete measure (defined for some non-Borel set). And note that in the case ...
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0answers
16 views

a question in calculus/real analysis

Can someone help me out to give a bound of the first derivative w.r.t. $x$ of the function $\min_{x,y\in\Omega}\{1,\frac{d(x,\partial\Omega)d(y,\partial\Omega)}{|x-y|^2}\}$?. ...
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1answer
28 views

prove product sigma algebra contains all open sets

I want to prove that $\mathcal{M} \times \mathcal{M}$ contains all open sets of $R^2$. I know that $\mathcal{M} \times \mathcal{M}$ is the product sigma algebra generated by the measurable rectangles ...
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21 views

Weak compactness of the unit ball of $L^p$

What does it mean by weak compactness of the unit ball of $L^{\frac{m}{m-1}}$?
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0answers
24 views

Measurable Function Proof Help

How to I prove that f is measurable if f is a real function of a measurable space $X$ such that $\{x ; f(x) \geq r\}$ is measurable for every rational $r$. From my basic understanding I need to prove ...
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0answers
17 views

On the importance of the Riesz–Markov–Kakutani representation theorem.

I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state ...
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0answers
23 views

Integrals of indicator functions question

I have a result $\int_X \int_Y \mathbb{1}[h(x,y) < \mu]dP(y)dP(x) < a$ and I am trying to resolve the integral $\int_X \int_Y \mathbb{1}[|f(x) - g(x)| > \frac{\mu}{2}] \mathbb{1}[h(x,y) ...
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2answers
103 views

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why and why isn't this true for $1/x$? I'm having difficulty understanding difference between the above functions in terms of ...
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0answers
41 views

Showing the expectation of uniform r.v is $\frac{a + b}{2} $

Suppose $X$ is uniform over $(a,b)$, then $\mathbb{E} \{ X \} = \frac{a + b}{2}$. I am given that $$ \mathbb{E} \{ X \} = \int X P^X(dx) $$ where $P^X$ is the distribution of $X$. I am confused ...
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0answers
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Question related to the definition of independent random variables

I have been given the following definition: Let $(X,M, \mu)$ be a probability space and let $F$ be a collection of function $f$ on $X$. The functions $f$ of $F$ are said to be "independent" if for ...
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0answers
16 views

Reference on Borel-Cantelli Lemmas [on hold]

I want some reference on Borel-Cantelli Lemmas. I know the usual two of them. There are many I guess. Thanks.
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1answer
17 views

What is the difference between algebra and $\sigma$-algebra generated in a finite space?

For a finite set $\Omega$ with (obviously finite) class $C$. Does the $\sigma$-algebra generated by $C$ $=$ the algebra generated by $C$? I think it does. But I'm not sure I understand if there is a ...
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1answer
33 views

Confused about switching Lebesgue integrals for Riemann integrals

Hi I have been attempting given in the link below. I am confused about the argument used to show the function is not Lebesgue integrable. This question What each person has used to answer is the ...
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0answers
14 views

Show that $\mathcal{B}_{\bar{\mathbb{R}}}$ is generated by $(r,\infty]$ for some $r\in \mathbb{Q}$.

I need to show that $\mathcal{B}_{\bar{\mathbb{R}}}$ is generated by $(r,\infty]$ for some $r\in \mathbb{Q}$. I know the definition of $\mathcal{B}_{\bar{\mathbb{R}}}$ which is ...
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0answers
26 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
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1answer
27 views

possibility of extension of Lesbegue measure on the real line

The Lesbegue measure on the real line is defined on the class of Lesbegue-measurable sets, whose elements satisfy the Caratheodory condition. Here what I am curious about is that if it is possible to ...
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1answer
13 views

Existence of a random variable given a cdf

For every real function F which can be a CDF (so has the properties that $F(+\infty)=1$, $F(-\infty)=0$, and F is non-decreasing and right continuous), does there exist a random variable on a ...
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1answer
11 views

$f$ is measurable iff $|f|$ is $s-$measurable. Show that $S=P(X)$.

Let $(X,S)$ be a measurable space such that for every function $f: X \to R$, $f$ is measurable iff $|f|$ is measurable. Show that $S=P(X)$. It is clear that $S \subset P(X)$. I need to show the ...
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0answers
17 views

for a given $f$, $f$ is measurable iff $f^{-1} (${$-\infty$}$) \in \mathcal{M}$ , $f^{-1} (${$\infty$}$) \in \mathcal{M}$ and f is measurable on $Y$.

Let $f : X \rightarrow \bar{\mathbb{R}}$ and $Y = f^{-1}(\mathbb{R})$ then f is measurable iff $f^{-1} (${$-\infty$}$) \in \mathcal{M}$ , $f^{-1} (${$\infty$}$) \in \mathcal{M}$ and f is measurable on ...
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0answers
10 views

Definition of a complex regular Borel measure

Let $X$ be locally compact Hausdorff space and let $\mathcal{B}$ be the Borel $\sigma$-algebra generated by the open subsets of $X$. I am familiar with the concept of a regular Borel measure $\mu$ ...
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1answer
16 views

Hankel transform with Bessel functions of the second kind

The Hankel transform is defined for Bessel functions of the first kind (see e.g. http://en.wikipedia.org/wiki/Hankel_transform) I would like to know if it is possible to define a Hankel transform ...
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1answer
8 views

If a Borel $\sigma$-algebra is genertated by a collection, then the collection is closed under intersections?

The question is: If $\sigma\left(\tau\right)$ is a Borel $\sigma$-algebra such that is genertated by a collection $\mathcal{B}\subseteq\sigma\left(\tau\right)$, then $\mathcal{B}$ is closed under ...
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1answer
21 views

¿If a Borel $\sigma$-algebra is generated by a collection of subsets of algebra, then the Borel $\sigma$-algebra is generated by the algebra?

Let $\left(M,\sigma\left(\tau\right)\right)$ a measure space with $\sigma\left(\tau\right)$ a Borel $\sigma$-algebra where $\tau$ is a topology in $M$. Suppose there is a algebra $\Gamma$ in $M$ ...
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1answer
46 views

Coin tossing probability (infinitely)

I just have viewed this question HERE My fist question is How the red rectangular is obtained from green one? Second, moving to infinite case: suppose $\sigma=\langle101\rangle$, let ...
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2answers
40 views

Proving that $m(E) = 0$ if for all $n$, $\int_E x^n \cos x\, d x = 0$

Suppose that $E\subset [0,2\pi]$ is measurable and $\int_E x^n \cos x\,dx = 0$ for all $n =0,1,2,\cdots$. Then prove that $m(E)=0$. In a non-rigorous fashion, if $\sum_{1}^{\infty} a_nx^n = \sec ...
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0answers
37 views

Connection between conditional and unconditional expectation

If $E[X\mid Y]=0$ a.s., then by the tower property of conditional expectation, $E[f(Y,z)X]=0$ for any integrate and measurable function $f$ and $z\in T$, where $T\subset\mathbb{R}$. Is it possible to ...
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1answer
75 views

Measure of an elementary set in terms of cardinality

In Terry Tao's textbook on measure theory and integration, he notes that, given an elementary set $A$, the length of $A$, denoted $|A|$, may be written discretely as $$|A| = \lim_{n \to ...
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probability measure on space of sequences

Let $\Omega=\{0,1\}^\infty$. For some $n$, let $B\subset \{0,1\}^n$. I have seen these two statements which make me confused little bit. (1) If $A\subset \Omega$, $A=B\times \{0,1\}^\infty$, and ...
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1answer
12 views

Distribution of a random measure is determined by the characteristic function

I ham trying to understand a proof from a book I am reading. It says the proof follows directly from the prior theorem and I just can't see that. Let $X$ be a random measure on a locally finite, ...
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0answers
28 views

If a simple function is measurable, then is its characteristic function

I am working on a problem looks like this: If a simple function $s$ is measurable, show that its characteristic function $\mathcal X_{X_i}$ is measurable. Here are the ways I have been working ...
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1answer
24 views

Approximation of Lebesgue-integrable functions by continuous functions.

I want to prove that if $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue-integrable, then for every $\epsilon >0$ there exists $g:\mathbb{R}\to\mathbb{R}$ continuous such that ...
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0answers
4 views

infinite intersection of jordan measurable sets

Is the infinite intersection of jordan measurable sets also jordan measurable? I´ve been trying to find a counterexample, but nothing so far. So is the statement true?
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2answers
21 views

Why does it hold $\operatorname{E}[Y\mid\mathcal{F}]=\operatorname{E}[Y\mid Y]=Y$, if $Y$ is $\mathcal{F}$-measurable?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathcal{F}\subseteq\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ $Y\in\mathcal{L}^1(\Omega,\mathcal{A},\operatorname{P})$ be ...
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1answer
62 views

If $\int f\;d\mu=\int g\;d\mu$, then $f\equiv g$ almost everywhere

I'm wondering whether or not the following statement is true: Let $(\Omega,\mathcal{A},\mu)$ be a measure space $f,g:\Omega\to\overline{\mathbb{R}}$ be measurable with respect to $\mathcal{A}$ and ...
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2answers
135 views

Understanding inverse of a function

I was trying to understand the proof for the following proposition. Proposition: If $\{f_n\}$ is a sequence of $\bar{\mathbb{R}}$ valued measurable functions on $(X,\mathcal{M})$, then the functions ...
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1answer
58 views
+50

Interpretation of measure theory question

I am struggling with the following question: Let $\Omega = \{1,2,3,4\}$, $\mathscr E = \{\{1\}\{1,2\}\}$ (a) Find $\sigma (\mathscr E)$ (sigma algebra) (b) By inventing an analog of a pmf or cdf, ...
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3answers
22 views

What is $\chi_{A_i}$

What is $\chi_{A_i}$ in eq(1)? I couldn't find the definition of it.
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2answers
76 views

Lebesgue Integral: $\int_1^{\infty}\frac{1}{x}$

The following is an exercise from Carothers' Real Analysis: Show that $$\int_{1}^{\infty}\frac{1}{x}=\infty$$ (as a Lebesgue Integral). Attempt: Let $E=[1,\infty)$. $\int_E f=\int f\cdot ...