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

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Symmetric borel sets in the plane

How will I show that the sigma algebra consisting of all Borel sets in the plane, which are symmetric about the line $y=x$, is generated by sets of the form $(a,b) \times (a,b)$? I could show upto the ...
2
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1answer
19 views

How to prove the following defined collection is a sigma algebra?

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
4
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2answers
107 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
0
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1answer
19 views

Probability densities and Absolute continuity

I've not deep knowledge in measure theory/real analysis but just few concepts given me during this second year probability course. I'm trying by myself to understand more, but I don't want to dive in ...
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0answers
29 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
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0answers
11 views

Folland's proof of the Hahn Decomposition. Minor error?

Theorem 3.3 of Folland's Real Analysis (ed 2) is the Hahn decomposition theorem. In the proof he assumes that the signed measure $\nu$ he is considering does not take the value $-\infty$. Then he ...
1
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1answer
18 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
4
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2answers
60 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
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1answer
65 views

Some properties of points with Lebesgue density equal to $1$.

I am studying Evans-Gariepy book and in corollary 1 of section 3.1.2, he prove that if $f:\mathbb{R}^N\to\mathbb{R}^M$ is locally Lipschitz and $$Z=\{x:\ f(x)=0\},$$ then $Df(x)=0$ a.e. $x\in Z$. He ...
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40 views

NUMBER OF ATOMS IN A SIGMA-ALGEBRA [on hold]

I have been trying to solve the followIng question. DESCRIBE THE SMALLEST SIGMA ALGEBRA CONTAINING 'n' ARBITRARY SUBSETS OF THE SAMPLE SPACE.GIVE AN UPPER BOUND FOR THE NUMBER OF SETS IN THIS SIGMA ...
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2answers
37 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
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1answer
39 views

sigmal algebra and measure [on hold]

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
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0answers
17 views

Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
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1answer
42 views

Unit ball in space of d dimension

If I have a unit ball in space $R^d$ then in how many dimension space its surface will be represented. I know the answer is d-1 but i am unable to convince myself. can anybody give me some intuition. ...
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1answer
18 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
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1answer
48 views

Question about statement of Fubini's theorem

This is a question on the statement of Fubini's theorem for measurable sets. The theorem looks like this: Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda = \mu \times \nu)$ be a complete ...
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2answers
136 views

Finding a “big” bounded subset of a subset of $\mathbb R$

I'm working in the following exercise. First, we define the outer Lebesgue measure, $m(A)=\inf\{\sum_n l(I_n): (I_n)\text{ is a sequence of open intervals and} A \subset \bigcup_nI_n \}$, where $l$ ...
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3answers
74 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
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1answer
39 views

The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
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1answer
284 views

Folland, Chapter 1 Problem 17

Problem 17: If $\mu^*$ is an outer measure on $X$ and $\{A_i\}_{i=1}^{\infty}$ is a sequence of disjoint $\mu^*$-measurable sets, then $\mu^*(E\cap \cup_{j=1}^{\infty} A_j)=\sum_{j=1}^{\infty}(E\cap ...
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0answers
28 views

Halmos Measure Theory section 39 Theorem D

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ...
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3answers
215 views

Regarding outer measure additivity [on hold]

I understand that outer measure is not countably additive. But if there are two disjoint sets, does the outer measure of their union equals the sum of their outer measure? Can someone give me a ...
3
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1answer
84 views

Almost all subgroups of a Lie group are free

I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions. Let $w\colon G\to H$ be an analytic mapping between ...
3
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1answer
35 views

If $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, does that imply $(X, \Sigma, \mu)$ is $\sigma$-finite?

I'm having trouble proving or disproving the statement: If the product space $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, then so is $(X, \Sigma, \mu)$. I ...
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1answer
46 views

Borel Measure on Banach Space

While thinking about what some measure on an infinite dimensional Banach space could look like a came across the point that if I'd like to assign a size to all epsilon balls, they by Riesz' lemma ...
1
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1answer
13 views

Limit distribution is invariant

Consider a homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with a countable (but not necessarily finite) state space $S$. Suppose that there exists a limit distribution $\pi$, namely: ...
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0answers
49 views

Explicit description of small open set containing the rationals

We know that the set $\mathbb{Q}$ of rational numbers has measure zero because it is countable. In fact, if $(q_n)_{n=1,2,\ldots}$ is an enumeration of $\mathbb{Q}$, then ...
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0answers
45 views

Lebesgue-integrable and existence of integral

I have given the following function $$ f(x,y) = \begin{cases} 1 &, \ x \in \mathbb{Q} \\ 2y & , \text{ otherwise} \end{cases} $$ This is a measurable function in sense of Lebesgue. Now, I ...
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0answers
31 views

Prove the following is a generated algebra

Let $ \mathbb E $ = {$A_{1}$, $A_{2}$,.......$A_{n}$} $\subseteq$ $\mathbb P (X)$ given, if we define $A ^a$ = \begin{cases} A & \text{if }a=0,\\ A^c & \text{if }a=1 \end{cases} for each $ ...
2
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1answer
64 views

Inequality of intervals lengths.

Let $(I_j)_{j = 1}^n$ be a finite collection of intervals that covers the rationals in $[0, 1]$. Prove that $\sum_{j = 1}^n \ell (I_j) \geq 1$. (Here, $\ell (I)$ denotes the length of the interval ...
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1answer
56 views

Some problems concerning regularity os measures.

Let $\mu$ be a regular "outer" measure on $\mathbb{R}^N$ (for example, the Lebesgue outer measure). By regularity I mean that for all $A\subset \mathbb{R}^N$, there is $E$ measurable with $A\subset E$ ...
2
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1answer
41 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
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1answer
376 views

How to prove that Lebesgue outer measure is translation invariant?

I am trying to prove that lesbegue outer measure is translation invariant, i.e., $m^\ast (E+y)=m^\ast E$. I proceed as follows. Let $E$ be a set. Let $\{I_n\}$ be a collection of open intervals that ...
2
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1answer
63 views

What exactly is a product measure?

If we have $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ a complete measure space with underlying complete spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$, and $\lambda = \mu \times \nu$, what ...
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1answer
19 views

Prove a set which conatins one point from each class in circle of circumference 1 is nonmeasureable

(Kolmogrov,p268,problem 7) Let C be a circle of circumference 1 and let $\alpha\in\mathbb R\setminus\mathbb Q$. Let all points f C which can be obtained from each other by rotating C through an ...
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1answer
78 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
3
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1answer
40 views

If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
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1answer
28 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
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1answer
31 views

If $f \in L^{1}(d\mu)$, is it true that $\int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu)$?

Ok, so we have $f \in L^{1}(d\mu)$, with $(X, \Sigma, \mu)$ a complete measure space. If we assume $f$ is nonnegative, we can define a measure $\rho(E) = \int \limits_{E} f \,d\mu$ for $E \in ...
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1answer
28 views

Minimum over measure

suppose I have a set defined by: \begin{align} A= \{ (x,y): \inf_z f(x,y,z) \le d \} \end{align} Now, if I define set: \begin{align} a(z)= \{ (x,y): f(x,y,z) \le d \} \end{align} Is it correct to ...
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92 views
+150

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
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1answer
25 views

Lebesgue measurable functions and the absolute value of them

Let $f$ is a measurable function. If $f$ is Lebesgue integrable, is the absolute value of $f$ Lebesgue integrable?
3
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1answer
45 views

Measure Theory Inequality

I was having trouble showing the following inequality: Prove that if $A \subset I = [0,1]$ has measure $u(A) < 1$ and $\epsilon > 0$, then there is an interval $[a,b] \subset I$ such that $u(A ...
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1answer
31 views

Integral $ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $ (Lebesgue)

I have to compute the following integral: $$ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $$ $\chi_{B_k}(x) =\begin{cases} 1 &, \text{if } x \in B_k \\ 0 ...
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0answers
22 views

Understanding the Definition of Lebesgue space (as it is given in a book of Katok/ Hasselblatt)

On page 733 of Modern Theory of Dynamical Systems by Anatole Katok and Boris Hasselblatt I found the following definition of Lebesgue space: Definition A.6.4. A measure space ...
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0answers
30 views

Derivation using Ito calculus?

I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let $L^{-}$ and $L^{+}$ be differential operators acting on ...
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1answer
28 views

If $E, F \subset [0, 1]$, $m(E), m(F) > 0$, and $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$, show $m(F\cap E_n) > 0$ for sufficiently large $n$

Suppose $E \subset [0, 1]$ has positive Lebesgue measure and let $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$. If $F \subset [0, 1]$ has positive Lebesgue measure, show that so does $F \cap E_n$ for ...
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0answers
36 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
0
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1answer
27 views

Ergodic (equivalent characterization)

Let $(\Omega,\mathcal{B},\mu,T)$ be a measuretheoretical dynamical system. Then this system is called ergodic if $$ B\in\mathcal{B}, T^{-1}(B)=B\implies \mu(B)=0\text{ or }\mu(B^C)=0. $$ ...
0
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2answers
87 views

If $f$ is a positive function and $\int_{E}f d\lambda = 0$ then $\lambda (E) = 0$

If $f$ is a positive function and $$ \int_{E}f d\lambda = 0, $$ then show that $\lambda (E) = 0$ where $\lambda $ is a Haar (Radon) measure. I know that if $f$ is a positive function and ...