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

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23 views

Defining Lebesgue measure on a subspace of $\mathbb{R}^n$

Let $\bar{w}_1,.., \bar{w}_k$ be linearly independent vectors in $\mathbb{R}^n$. Let $W$ be the subspace spanned by these $\bar{w}_i$'s. I know how the Lebesgue measure is defined on $\mathbb{R}^n$. ...
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0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
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1answer
16 views

probability of tail event using kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
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1answer
36 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
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0answers
10 views

For sequence of events $\{A_n\}$ in probability space, show that $\lim_n P( \lim\inf_k A_n \cap A_k^c)=0$ [on hold]

I think it has to be done by considering $A_n$ and $A_k^c$ separately.
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1answer
18 views

Example of strict inequality in special case of fatou's lemma.

Give an example of sequence of events $\{A_n\}$ such that the following inequalities are strict $P(\lim\inf A_n) \le \lim\inf P(A_n) \le \lim\sup P(A_n) \le P(\lim\sup A_n)$. Thanks
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1answer
9 views

Borel Measurable Function 3

So the definition of a measurable function is as follows: Let $f: (X, \mathcal{A}) \rightarrow\overline{\mathbb{R}}$ be a function on the measurable space $(X, \mathcal{A})$. Then $f$ is said to be ...
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0answers
15 views

Weak Compactness Therem for $L^p$

I have a problem to understand a point in the Weak Compactness Theorem from the book of Evans/Gariepy. So we have $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ a Radon measure with $\mu \ll ...
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1answer
21 views

Open Sets in the Extended Real Line

So I know that the extended real line is given by $\mathbb{R} \bigcup$ {$-\infty, \infty$}. So these are the facts that I know: 1) Firstly, every interval in $\mathbb{R}$ is a Borel Set (I seem to ...
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0answers
26 views

Borel $\sigma$-algebra

Since the Borel $\sigma$-algebra is generated by the family of open sets, does that mean that every Borel set is essentially some countable union/intersection of open sets or a complement of open ...
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2answers
18 views

Borel Sigma Algebra generated by Open Intervals

So I know that the Borel $\sigma$-algebra of $\mathbb{R}$ is the $\sigma$-algebra generated by open sets. I have been able to prove that this Borel $\sigma$-algebra is also generated by the family of ...
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0answers
27 views

Marking Integers Using a Wheel

Suppose I had a wheel of diameter one meter and I was charged with marking every meter along an infinite stretch of a beach. The strategy is to insert pegs into the wheel so that every point that is a ...
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1answer
32 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable? [on hold]

If the answer is yes, how to prove that? Otherwise how to find a counterexample?
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1answer
50 views

Prove Y = X given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$

Prove Y = X, given $Y = E[X|\mathscr{G}] $ and $EY^2 = EX^2$ Attempt: Suppose $Y = E[X|\mathscr{G}] $. Then $E[X|\mathscr{G}] $ is $\mathscr{G}$-measureable. For every A $\in \mathscr{G}$: ...
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1answer
92 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
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1answer
28 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
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1answer
26 views

Prove a sequence of integrals converges to 0

Let $E$ be a set of finite Lebesgue measure in ${\bf R}$ and $\{a_n\}_{n \in {\bf N}}$ be a sequence of real numbers. Show that $\int_E \cos(nx + a_n) dx $ goes to 0 as $n \to \infty$. I tried ...
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1answer
22 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ ...
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1answer
23 views

Integrability of dirichlet function in $\mathbb{R}^3$

Let $d: [0,1] \rightarrow \mathbb{R}$ be the Dirichlet function as follows: $$d(x) = \begin{cases} 1, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{R} \backslash \mathbb{Q} ...
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0answers
15 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
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0answers
16 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
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1answer
14 views

Finite measure space & sigma-finite measure space

A measure space $(X, \Sigma, \mu)$ is finite if $\mu(X)<\infty$. It is equivalent to saying that $(X, \Sigma, \mu)$ is finite if $\mu(E)<\infty$ for all $E \in \Sigma$ A measure space $(X, ...
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0answers
7 views

Weil's definiton of image of Haar measure on homogeneous space $G/\Gamma$ where $\Gamma$ is discrete

Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. Let $\Gamma$ be a discrete subgroup of $G$, $G/\Gamma$ the ...
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1answer
37 views

Finite meaure space with $f \in L^p$ [duplicate]

Given a finite measure space $(X,\Sigma,\mu)$, for $1<p<\infty$, if $f \in L^p(X)$, then $f \in L^1(X)$. Can anyone show me how to start the proof? Thanks.
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1answer
30 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
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1answer
17 views

Demonstration that pure point set is countable if the measure is finite for every compact

I read on Reed and Simons' this statement. Let P be the pure point set of a positive Borel measure $\mu$ on $\mathbb{R}$, that is $P = \{ x\, |\, \mu(\{x\}) > 0 \}$. Then this set is countable if ...
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2answers
53 views

Is a subspace of functions that essentially depend only on one variable closed?

Let $S$ be the subspace $$\left\{f\in L^p( I^2)|\exists g\in L^p( I), f(x,y)=g(x), \mbox{a.e. } (x,y)\in I^2\right\}.$$ Is $S$ closed under the $L^p$ norm? I think the first step would be to ...
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2answers
252 views

Lebesgue non-measurable function

Can we give an example of Lebesgue non-measurable function, for which set $\{x: f(x)=C\}~\forall C\in\mathbb{R}$ is measurable? Thanks.
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1answer
21 views

Borel Sigma Algebra- Measure Theory

I have tried looking at various sources and still cant understand the following: Consider $X= \mathbb{R}$, then the $\sigma$-algebra generated by the family of closed intervals $[a,b]$ is the same as ...
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0answers
9 views

Measure of convolution

Let $M$ be the Banach space of all complex Borel measures on $R$.The norm in $M$ is $\|\mu\|=|\mu|(R)$,associate to each Borel set $E\subset R$ the set $$ E_2=\{(x,y):x+y\in E\}\subset R^2 $$ if ...
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0answers
14 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
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2answers
32 views

Borel $\sigma$ -algebra

So I have a proposition which states the following: Each of the following families of sets generate the Borel $\sigma$-algebra: 1)The family of all open intervals $(a,b)$, $a,b, \in \mathbb{R}$. 2) ...
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1answer
34 views

Is the following statement true on $L^0$ spaces?

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X,Y\in L^0(\Omega;\mathbb{R})$ two random variables taking values in $\mathbb{R}$. Is it true that: $$\int_{A} f(X(\omega)) P(d\omega) = ...
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1answer
18 views

Borel Sets- Measure Theory

Let $X$ be a metric space. Then the family of Borel Sets in $X$ is the $\sigma$-algebra generated by the family of open sets. So if I am not mistaken are we saying that, consider $X$ to be any ...
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1answer
23 views

Almost uniform convergence implies a.e. pointwise convergence proof

I've just read a proof of the statement "On a finite measurable space, $(f_n)_{n \geq 1}$ and $f$ measurable and finite a.e. functions, if $(f_n)_{n \geq 1}$ converges almost uniformly to $f$, then it ...
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2answers
32 views

Hölder inequality conditions for $L_p$ spaces?

The Hölder inequality is the statement that if $f,g$ are measurable functions then $$ \|fg \|_1 \le \|f\|_p \|g\|_q$$ if $p,q$ are such that ${1\over p}+ {1 \over q} =1$. But it's not clear to me ...
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0answers
18 views

Explicit construction of Haar measure on quotient group

Let $G$ be a locally compact Hausdorff group, $H$ a closed normal subgroup. To simplify matters we assume that the underlying topological space of $G$ has a countable base. Suppose a left Harr ...
2
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1answer
55 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
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1answer
16 views

Integrating with respect to a linear combination of two signed measures

Let $(X, d)$ be a metric space and $\mathcal{B}(X)$ the Borel $\sigma$-algebra of X. Let $\mu, \nu$ be two real-valued signed measures defined on $(X, \mathcal{B}(X))$ and $f : X \to \mathbb{R}$ Borel ...
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2answers
36 views

Lusin's theorem clarification

I am looking at lusin's theorem proof in rudin's book and the only detail I cannot understand is when he says that $2^{n}t_{n}$ is the characteristic of some subset $T_{n}$. (3rd line of the proof) ...
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1answer
12 views

intuitive proof regarding measure of single element set on a continuous distribution

I'm brushing up on measure theory and was hoping to get a somewhat intuitive explanation of the following. Suppose that $\Omega = [0, \infty)$ and our $\sigma$-algebra $A = \mathcal{B}([0,\infty))$ ...
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0answers
18 views

Good partition in locally compact metric space

Let $X$ be a locally compact second countable metric space, endowed with a Borel probability space $\mu$. Let $\varepsilon>0$. Question: Is it possible to find a countable Borel partition $\xi$ of ...
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0answers
32 views

Measurability of the set of all differentiable points

Given a function $f:(a,b)\to\mathbb{R}$, is the set of all points where $f$ is differentiable a measurable set? Please provide a short proof or a counterexample. And furthermore, in case it's false, ...
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3answers
43 views

Can I cover a square with many line segments?

Not sure If I've chosen the tags correctly. Anyway, is it possible to obtain a unit square with enough line segments oriented vertically, placed next to each other? We know that a unit square has ...
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0answers
22 views

MCT from Fatou's Lemma [different proof]

Assume $\forall \left\{ f_n \right\} \subset L^+$ we have $\int \liminf f_n \leq \liminf \int f_n$. Now prove MCT: If $f_n\nearrow f\in L^+$ then $\lim \int f_n=\int f$. I have seen several suggested ...
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1answer
31 views

Process adapted to Filtration [on hold]

Here is the definition I have been given : A process $(X_t)$ is adapted to a filtration $(\mathcal F_t)$ if $X_t$ is $F_t$ measurable, for all t > 0 , i.e : $X_t^-1 (\mathcal B)$ belong to ...
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1answer
17 views

Outer Measure on a Probability Space is 1 iff its complement is null?

I am trying to prove the following theorem, which I feel should be true, but am not sure how to go about it. Suppose we are in a probability space $(\Omega, \mathcal{F}, P)$ and we define for any ...
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2answers
32 views

Real function such that preimage of every constant is measurable

Let $f:\mathbb R \to \mathbb R$ such that for each $c \in \mathbb R$, the set $\{x \in \mathbb R:f(x)=c\}$ is measurable (Lebesgue measurable), is $f$ measurable? I've came up with a solution and I ...
1
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1answer
46 views

Conditioning events on a conditional expectation

Let $X_0=a$ for some $0<a<1$ and for $n \geq 0$, let $\mathbb{P}(X_{n+1}=X_n/2|\mathcal{F}_n)=1-X_n$ and $\mathbb{P}(X_{n+1}=(1+X_n)/2| \mathcal{F}_n)=X_n$. Show that X is a martingale. This ...
1
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1answer
24 views

Is $\operatorname{card}(I)=\operatorname{card}(D)$

When I was answering number of integrable functions is greater than number of differentiable functions I got to wonder if the inequality was strict. So with $\mathcal I$ being the set of integrable ...