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

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407 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
5
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1answer
306 views

Prove Sine integral exists as improper Riemann integral but is not Lebesgue-integrable.

I got to prove that $$\int_0^1 \frac{1}{t}\sin\left(\frac{1}{t}\right)dt,$$ exists as an improper Riemann integral, yet that $$f(t)=\frac{1}{t}\sin\left(\frac{1}{t}\right)\notin ...
2
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0answers
179 views

Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by ...
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1answer
58 views

Is $f =g$ when $g=\limsup f_n$?

f and g are two functions . Is $f =g$ $\mu$ a.e when $g=\limsup f_n$ when n -> infinity ? we have $f_n$ --> f $\mu$ a.e , $f_n$ is measurable for all n in N. I think they will equal because when ...
4
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1answer
233 views

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
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1answer
39 views

what does $\varnothing$ map to in a measurable function?

Here is the question: I am trying to prove a proposition from Schervish. Suppose that $(S,\mathcal{A})$ and $(T,\mathcal{C})$ are measurable spaces. $f:S\rightarrow T$ is a function. If ...
2
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2answers
368 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
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2answers
284 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 ...
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0answers
85 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
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1answer
38 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
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2answers
70 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
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1answer
172 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
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0answers
48 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
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1answer
53 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
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1answer
38 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
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1answer
37 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
101 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 ...
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2answers
97 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 ...
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2answers
106 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 $k \lt n$, ...
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2answers
395 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
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0answers
73 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
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1answer
171 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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441 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, ...
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2answers
104 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 ...
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1answer
212 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
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1answer
68 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: ...
4
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0answers
235 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 ...
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2answers
54 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
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1answer
719 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 ...
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1answer
82 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
132 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
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1answer
108 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
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1answer
50 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
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1answer
54 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 ...
1
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2answers
116 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 ...
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1answer
86 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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2answers
972 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
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0answers
91 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 ...
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2answers
119 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
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1answer
224 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) = ...
3
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2answers
219 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
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1answer
70 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
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1answer
68 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
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1answer
569 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 $f_{n}(x)=n^{-1}\chi_{(0,n)}$ ...
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1answer
140 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 ...
3
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2answers
559 views

Bepo - levi implies monotonone convergence theorem??

For Lebesgue measurable functions, Is is true that Beppo-Levi theorem implies the monotone convergence theorem? Beppo-Levi: Suppose $ \sum_{k=1}^{\infty} \int |f_k |\, dm $ is finite. Then the series ...
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2answers
110 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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2answers
222 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
46 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 ...
4
votes
1answer
486 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 ...