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

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44 views

Does the integral converge I can't find counterexample

I found the following question in the book of kolomogorov fomin introductory real analysis and I don't know how to solve it. Does anyone have any ideas? Suppose $f$ is integrable on sets ...
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1answer
24 views

Measurability of function

Let $g: C[0,\infty) \to [0,\infty)$ be a Borel-measurable function. Define $f: C[0,\infty) \times C_0[0,\infty) \to C[0,\infty)$ by $f(y,z)(t)=y(t)+z(1-g(y))\mathbf{1}_{\{t > g(y)\}}$. In a proof ...
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0answers
47 views

proof that Riemann integrals is extended by Lebesgue integrals

After reading a proof sketch somewhere (forgot the link) I've written a proof in my own words. I'm not quite sure if I got the details right, since there were variants of this floating around that any ...
4
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1answer
55 views

A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
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0answers
133 views

Joint Expectation of independent Random Variables given two sigma-algebras

We have a question regarding two random variables $X$,$Y$ on a probability space with sigma-algebra $\mathcal{F}$ and a sub-sigma algebra $\mathcal{M}$ such that $X$ is independent of $\mathcal{M}$ ...
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1answer
83 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
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0answers
56 views

Nonzero Radon-Nikodym derivative invertible?

Suppose that $\nu$ is a $\sigma$-finite positive measure, and that $\rho$ is a measurable function that's nonzero $\nu$-a.e. Define $\mu(A) = \int_{A} \rho\, d\nu$ for all $\nu$-measurable $A$, so ...
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2answers
117 views

Do Riemann and Lebesgue integrals always agree?

I know that on a closed bounded interval, say $[a,b]$ in $R^1$, if a function is Riemann integral, then it is Lebesgue integrable, and the values of those two integrals are the same. But, is this ...
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1answer
28 views

Is it right to say that a positive measure is a signed measure by definition?

A signed measure $\mu$ is a measure which can also take on negative values. Now my question is, is a positive measure a special case of a signed measure since it essentially maps to a subset of ...
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1answer
80 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
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1answer
112 views

Basic finite dimensional distribution question

I'm having trouble wrapping my head around the basic idea of a finite dimensional distribution. Suppose $(\Omega, \Bbb P, \mathcal{F})$ is a probability space. Let $(X_{t})_{t \geq 0}$ be a ...
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1answer
52 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
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0answers
41 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
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1answer
91 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
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0answers
48 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
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0answers
42 views

Wiener measure of smooth function in space of continuous function.

How do we show that the Wiener measure of class of smooth functions in $C[0, \infty)$ is 1?
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1answer
54 views

Lebesgue measure without choice

From this question and this question (and their answers) I gather that it is consistent with ZF without The Axiom of Choice to assume that there exist countable sets $A_n$, $n\in \mathbb N$, such that ...
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0answers
69 views

Why the two expressions of total variation distance are equivalent?

In a stochastic processes textbook, I find the definition of total variation distance is $\|\pi - \nu\|_{TV} = \max\{|\pi(A) - \nu(A)|:A\subset S\}$ where $\pi$ and $\nu$ are two probability measures ...
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1answer
607 views

Why does a Borel measurable function imply its Lebesgue measure?

Borel measurable defined as: $f: D ->\mathbb R$ is Borel measurable if $D$ is a Borel set and if, for each real $a$, the set {$x∈D: f(x) > a$} is a Borel set. Definition of Lebesgue measurable ...
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1answer
27 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
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2answers
43 views

$f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.

Let $X=(\mathcal{X},\mathcal{M},\mu) $ be a measure space. Assume that $\mu$ is $\sigma$ finite and $1\leq p \leq \infty$, with $q$ the Holder conjugate exponent. If $f_1,f_2 \in L^q(\mu)$ and ...
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1answer
61 views

On the Preservation of Product Measurability under Partial Conditional Expectation.

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces, $\mathcal{X}_{0}\subset\mathcal{X}$ a (sub)sigma field and assume that $f=f(x,y)\in L^{1}_{\mu\otimes \nu}$ where $(X\times ...
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0answers
163 views

How to learn problem solving strategy for Measure Theory?

I have taken both graduate level Algebra and Measure theory courses but found the latter much more difficult for me. I have put a lot effort on learning it by reading a few reference books and ...
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1answer
81 views

Approximation of one function by other using a smooth multiplier function

This problem is from the Book, Harmonic Analysis by Katznelson (Problem 2, Page 160). Suppose $f$, $g\in L^2(\mathbb{R})$ such that $f(x) = 0$ implies $g(x)=0$ for almost all $x\in\mathbb{R}$. Then ...
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2answers
119 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
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1answer
40 views

Outer Measures - Measure Theory

In the definition of an outer measure, they state the sub-additivity condition as $\mu_{*}(\bigcup A_{n}) \leq \sum\mu_{*}(A_{n})$ for any sequence of sets $A_{n} \subset X$ My question is does ...
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1answer
106 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
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2answers
100 views

Lebesgue measure of graph of $\sin{\frac{1}{x}}$ on $[0,1]$

I am working on something and read that measure of graph of a continuous function on compact sets is zero. Now, I tried to do it for non continuous functions but the set of discontinuities have ...
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3answers
138 views

If $K\subset\mathbb{R}$ is compact and has positive Lebesgue measure, is the interior of $K$ non-empty?

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
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2answers
97 views

Closed subgroup of a locally compact Hausdorff group whose Haar measure is non-zero.

Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$. ...
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0answers
60 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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1answer
145 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
82 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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1answer
71 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
25 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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1answer
147 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
65 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
235 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
41 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
125 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable?

If the answer is yes, how to prove that? Otherwise how to find a counterexample? Update: I've figured out the tricks inside. A countable intersection of open sets in $\mathbb R$ is equivalent to a ...
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1answer
253 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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141 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
51 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
36 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
76 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
37 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
28 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
31 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
64 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
51 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.