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

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2
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2answers
221 views

Why do we call it a $\sigma$-algebra?

In simple terms, a $\sigma$-algebra is the collection of all of the things we know how to measure. Why don't we call it something that more directly suggests this, for example a 'measure space?'
2
votes
2answers
171 views

Convergence almost everywhere and convergence in measure

Let $(\mathbb{R},\mathcal{L},m)$, let $f_{n}(x)=n\chi_{[0,\frac{1}{n}]}$ then the sequence converges to $0$ everywhere except at $x=0$ thus $f_{n}$ converges a.e. Then in my book (Folland) we have ...
1
vote
0answers
63 views

Borel $\sigma$-field in k is same as product Borel $\sigma$-field using open balls

I am supposed to prove "The Borel $\sigma$-field $\mathcal{B}^k$ of $\mathbb{R}^k$ is the same as the product $\sigma$-field of k copies of $(\mathbb{R},\mathcal{B}^1)$." I have already done the ...
0
votes
1answer
34 views

Lebesgue-Stieljes measure and substitution counterexample.

The goal of this problem is to show that in general, $\int_{(0,\infty)} f(x) d\mu(x) \neq \int_{(\alpha,\infty)} f(x-\alpha) d\mu$. This was an assumption I made on a math problem which led me to ...
2
votes
2answers
65 views

Limit of a Lebesgue integral

What is the value of: $$\lim_{n\to\infty}\sqrt{n}\int_0^{1}(1-t^2)^ndt$$ I think I have to use the Theorem of dominated convergence
-1
votes
1answer
107 views

Chebyshev inequality- Conclusion

Let $(X,\Omega,\mu)$ be a measure space and let $f$ be an extended real valued measurable function defined on $X$. I have already seen that $$ \mu\left(\{x\in X : |f(x)|\geq t\}\right)\leq ...
2
votes
0answers
43 views

Convergence of integral under a limit

Background: Let $X$ and $Y$ be two lognormal random variables, and $Z = X|Y = y$ a lognormal random variable obtained by conditioning on $Y$. Denote by $g_{\rho}(z)$ the probability density function ...
2
votes
1answer
46 views

Chebysev inequality proof

Let $(X,\Omega,\mu)$ be a measure space and let $f$ be an extended real valued measurable function defined on $X$. Proof that $$ \mu\left(\{x\in X : |f(x)|\geq t\}\right)\leq \frac{1}{t}\int_X ...
1
vote
1answer
24 views

smallest algebra generated by ring

Let $\Omega\neq \emptyset$ and $\mathcal E$ be a ring in $\Omega$ and $\mathcal R_0 := \mathcal E\cup \{E^c:E\in\mathcal E\}$. Show that $\mathcal R_0$ is the smallest algebra in $\Omega$ containing ...
2
votes
1answer
35 views

Can we replace the condition $d(E_1,E_2)>0$ with $E_1\cap{E_2}=\emptyset$ in an property of outer measure?

Can we replace the condition $d(E_1,E_2)>0$ with $E_1\cap{E_2}=\emptyset$ in "If $E=E_1\cup{E_2}$,and $d(E_1,E_2)>0$,then $m_\star(E)=m_\star(E_1)+m_\star(E_2)$."? ...
2
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0answers
98 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
0
votes
2answers
94 views

Differentiability at $0$

Let $\phi:[0,1]\to\mathbb R$ be a Lebesgue measurable and integrable function. Consider: $$F:t\in [0,\infty ) \to \int_0^1 \sqrt{t+\phi^2(x)}dx.$$ What is a necessary and sufficient condition for the ...
1
vote
2answers
85 views

Question About Lipschitz Maps and Measure Zero

I'm having difficulty with a problem involving measure. Suppose $F : B^k \rightarrow B^n$ is a Lipschitz map from the unit ball in $\mathbb{R}^k$ to the unit ball in $\mathbb{R}^n$. If $k \lt n$, ...
3
votes
2answers
243 views

Proving completeness of Nikodym Metric

I'm trying to prove completeness directly of the metric given by $d(A, B) = \mu (A \triangle B)$ on a finite measure space $(X, M, \mu)$. Edit: I should make clear that I'm referring to completeness ...
2
votes
0answers
48 views

Proving product measures: Homework

Being an engineer I'm at loss how to prove the following exercises, and I would appreciate any comments. Prove Fubini's theorem for an $\mathcal{L}^1$ integrable function $f$. Here's my stab at it. ...
2
votes
1answer
98 views

Continuously differentiable curves

How one can show that the unit cube $[0,1]^d$ cannot be covered by countably many continuously differentiable curves? Thanks in advance
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0answers
232 views

What is the Dirac mass on measure space?

I am reading the book "Lectures on Stochastic Analysis." But I know seldom about measure space. I meet with a symbol which the author call Dirac mass(in 9.3 of this book). Let E be a measurable space, ...
0
votes
2answers
98 views

$f = 0 $ almost everywhere implies $\int_{\mathbb{R}} f = 0 $

my try: Suppose $\phi $ is imple function such that $ 0 \leq \phi \leq f$, then since $ f = 0 $ almost everywhere, then $\phi = 0 $ almost everywhere. Let $A = \{ x : \phi(x) = 0\}$. Therefore, $A^c ...
1
vote
1answer
163 views

Symmetric difference of sets and convergence in integration.

Let $(X,\mathcal{M},m)$ be a space of measure and $f_n,f \in L^1(m)$ such as $||f_n - f||_1 \rightarrow 0.$ Suppose that we also have $A_n,A \in \mathcal{M}$ and $m(A_n \triangle A) \rightarrow 0.$ ...
0
votes
1answer
51 views

what does eventually mean in the following question ? thanks

Suppose $\phi < f $, $g_n \to f $ pointwise, $g_n = inf_{k \geq n} f_k $. MY books says that 'eventually' $g_n \geq \phi$. What do they mean by eventually? Also to show this, they do the following: ...
2
votes
0answers
136 views

Please help in Folland Analysis Proposition 2.11

I don't really understand proposition 2.11 in Folland. So please help me to explain, as well as give some hints to prove it. The part makes me confuse is $f=g$ $\mu-a.e$ ,does it means f equals to g ...
2
votes
2answers
50 views

Clarification on HW Question: Show $inf_{n}P(A_{n})>0$ implies $P(A_{n}\text{i.o.})>0$.

So this was a problem on one of my assignments this week in an introductory probability course, and am a bit confused because I think I am mis-interpreting what the notation means and what the point ...
2
votes
1answer
460 views

Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
0
votes
1answer
58 views

$p\mapsto\Vert f\Vert_{L^p}$ is continuous. [duplicate]

Could someone help me prove the following: Let $(\Omega,\mathscr{A},\mu)$ be a probability space. Let $f:\Omega\rightarrow\mathbb{R}$ be a non-negative measurable function. How do I prove that the ...
3
votes
2answers
104 views

Ergodic for the mean, but not ergodic stochastic process?

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean ...
1
vote
1answer
85 views

Convergence of Integrals of sequence of functions

Let $f_{n}$ be a sequence of integrable functions on $R^d$ that converges to $f$ a.e., and assume there exists a number $c$ such that for all $n$, $\int|f_{n}(x)| <c$. Prove that $$\lim_{n ...
0
votes
1answer
40 views

Reparametrization of an absolutely continuous curve

If $\alpha : [0,1] \rightarrow \mathbb{R^n} $ is $C^1$ and $\alpha'(t) \neq 0$ for all $t\in[0,1]$ then there always exists a reparametrization in which $\| \alpha'(s) \| = 1$. Is there an equivalent ...
0
votes
1answer
48 views

Meaning of symbol $L^1(\mathbb{P})$

In Furstenberg-Kesten theorem, a theory relating to products of random matrices, one of the assumptions is that: $$\log^{+}||A||\in L^1(\mathbb{P}),$$ where $A$ (a random matrix) is the generator ...
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2answers
96 views

cantor middle thirds set

Let $f: [0,1] \to \mathbb{R}$ be defined by letting $f = 0 $ on $\mathcal{C}$, the Cantor set and $f(x) = k $ for every $x$ in each interval of lenght $\frac{1}{3^k}$ which has been removed from ...
0
votes
1answer
59 views

$\sigma$-algebras, containment, and the notation used.

I'm proving the following proposition taken from Royden 4th Edition, Let $\mathcal{F}$ be a collection of subsets of a set $X$. Then the intersection $\mathcal{A}$ of all $\sigma$-algebras of subsets ...
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vote
2answers
516 views

Sequence of simple functions nonnegative that converge to measurable function $f$

Suppose $f\geq 0$ is measurable. We want to find a sequence of $s_n$ of nonnegative simple functions such that $s_n \to_{pointwise} f$. My book says the we should consided the sequence: $$ s_n = ...
3
votes
0answers
79 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
1
vote
2answers
87 views

Borel sets defined for closed sets

This is adapted from 1.7.7 in Friedman's "Foundations of Modern Analysis": Let $\mathscr{B}$ be the $\sigma$-ring generated by the class of open subsets of $X$ [a fixed set], and $\mathscr{D}$ the ...
4
votes
1answer
126 views

Stable Convergence in Distribution - Martingale CLT problem (Lemma 3.1 in Hall and Heyde)

I'm studying Hall and Heyde's (1980) book on martingale limit theory. In their Lemma 3.1, they seem to use the identity \begin{equation} \mathrm{E}\left({\exp{(itZ)}\mathbb{1}_A}\right) = ...
2
votes
2answers
199 views

Measure of the set of all $x$ such that $A-x$ contains infinitely many integers

Let $A$ have finite Lebesgue measure and let $B$ be the set of all $x$ in $\mathbb R$ such that $A-x$ contains infinitely many integers. Prove that $m(B) = 0$. As soon as I saw infinitely many, I ...
2
votes
1answer
52 views

Clopen and open sets have the same measure

I was positive I had already asked this one but apparently not. It is simply this: Show that the Lebesgue outer measure of $[a,b)$ is $b-a$. Let $\mathscr{K}$ be the collection of open subsets ...
2
votes
1answer
62 views

Points of Convergence (functions not assumed to be non-negative)

I am interested in a modified version of this question. Namely, let $\left\{f_n(x)\right\}$ be a sequence of continuous functions on $[a,b]$. Prove that : S = $\left\{x \in [a,b] ...
2
votes
1answer
360 views

almost everywhere convergence vs uniform convergence

Let $(\mathbb{R},\mathcal{L},m)$ Can someone explain to me why $f_{n}(x)=\chi_{(0,\frac{1}{n}]}$ converges almost everywhere to $0$ but not uniformly... also why does $f_{n}(x)=n^{-1}\chi_{(0,n)}$ ...
0
votes
1answer
79 views

How is the Borel-Cantelli lemma used in this proof on $\psi$-approximable numbers?

I'm trying to understand a paper called "Almost no points on a Cantor set are very well approximable". In the proof the author uses the Borel-Cantelli Lemma (in the eighth line at the beginning of the ...
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2answers
101 views

how to prove $E(X|\mathcal G)(\omega)=n\int^{\frac{j}{n}}_{\frac{j-1}{n}}X(s)ds, \omega\in(\frac{j-1}{n},\frac{j}{n}].$ [closed]

Suppose $\Omega=[0,1]$, and $\mathcal P=$lebesgue Measure , and $\mathcal F=\mathcal B([0,1])$ and also Suppose X is random variable and $\mathcal G$ is $\sigma-$algebra Produced With intervals $ ...
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votes
2answers
157 views

$\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$

I want to show that $\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$. For example, take $n=1$. I want to show that the $x$-axis has measure zero in the plane. I cover it with the sets ...
2
votes
0answers
31 views

Scheffe's theorem

Scheffe's theorem states that consider the real line $\mathbb{R}$ and sequence of probability densities $\{f_n\}$ such that $f_n \to f$ pointwise. Then if $\{P_n\}$ and $P$ denote the reps. measures ...
2
votes
1answer
241 views

On functions whose derivative equals zero almost everywhere

Suppose $f: [0,1] \rightarrow \mathbb{R}$ is continuous everywhere and differentiable almost everywhere in $[0,1]$, and $f'(x)=0$ whenever the derivative exists. Is it true that $f(x)$ equals a ...
0
votes
1answer
61 views

Prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.

If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable. My thoughts: We know for any sequence ...
2
votes
1answer
86 views

Showing that a Sequence Containing Indicator Functions Converges a.s.

Let $f_{n}(x) = n1_{[0,1/n]}(x)$, where $f_{n}$ maps R with the Borel sigma-algebra to itself. Also, let $\Omega = [0,1]$, your sigma-algebra is the Borel sets in [0,1], and $P(dx)=dx$. Show that the ...
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1answer
169 views

Switching the Order of Summations

Under what conditions is it okay to switch the order of two nested infinite summations? I am working on a proof in order to show that a particular set function is a probability measure. In order to ...
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votes
1answer
56 views

Does $\bigcup _{j\in\mathbb{N}}A_j=A$

I'm reading R. Schilling's Measure, Integrals and Martingales and in a proof he makes the following statement. "Since $A=\bigcup _{j\in\mathbb{N}}A_j$...", is this allways true? The context: The ...
2
votes
1answer
192 views

The definition of Borel sigma algebra

In the text of Probability Essentials by J.Jacod & P.Protter, a theorem: The Borel $ \sigma $- algebra of $R $ is generated by intervals of the form $(-\infty,a ]$, where $a \in Q$. As far as ...
1
vote
1answer
57 views

why the volume of $Q$ is equal to the volume of the closure of $Q$? Where $Q$ is an open cube.

Given an open cube $Q\subset \mathbb R^n$, $\overline{Q}=Q\cup \alpha Q$ where $\overline{Q}$ is the closure of $Q$ and $\alpha Q$ is the boundary of $Q$. I know $|\overline{Q}| =|Q|$,where $|Q|$ is ...
4
votes
1answer
175 views

Prove that the $\sigma$ - algebras are equal

I want to show that $\sigma$-algebras on $\mathbb{R}$ generated by $(a,b), \ (a,b], [a, b), [a,b], (-\infty, a), (-\infty, a], (b, +\infty), [b, +\infty)$ for $a,b \in \mathbb{R}$ and $a,b \in ...