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

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1answer
27 views

Wrong proof: The outer measure of [0,1]= 0

The definition of outer measure of a set $E$ in $\mathbf{R}$ is: $m_*(E)=inf\Sigma|I_j|$ where ${I_j}$ the family of open intervals that cover $E$. I am wondering why this proof is wrong: $\forall ...
0
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0answers
8 views

Treves definition of the upper integral of an arbitrary non-negative function

Let $(X, \Sigma, \mu)$ be a mesure space where $\Sigma$ is a $\sigma$-algebra on a set $X$ and $\mu$ is a measure defined on $\Sigma$. We assume $\mu$ is $\sigma$-fimite, i.e. $X$ is a countable union ...
-1
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0answers
4 views

f(x) Lebesgue measurable and integrable, show g(x,y) = f(x)/x is wrt product measure

f(x) Lebesgue measurable and integrable, show g(x,y) measurable and integrable is wrt Lebesgue product measure on (0,1 x (0,1) where g(x,y) = f(x)/x for 0< y< x<1 and 0 elsewhere
0
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0answers
12 views

The finite-dimensional distributions of a centered Gaussian process are uniquely determined by the covariance function

Let $I\subseteq\mathbb{R}$ and $X=(X_t)_{t\in I}$ be a centered Gaussian process, i.e. - $E[X_t]=0$ for all $t\ge 0$ - $X$ is real-valued and for all $n\in\mathbb{N}$ and $t_1,\ldots,t_n\ge 0$ we've ...
-1
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0answers
8 views

Some questions on convergence of measurable functions and increasing sequences.

My question relates to the proof of existence of the essential supremum, on Planetmath (http://planetmath.org/?op=getobj&id=11400&from=objects). I have some qualms with the proof so would ...
0
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1answer
14 views

Question on finite dimensional distribution of Markov Chain

If $\{ X_{n} \}$ is a Markov Chain and $X_{o} \sim \pi$ (where $\pi$ is the stationary measure), it follows that the MC is identically distributed. I have a question about the finite dimensional ...
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0answers
18 views

A dense subset in $L^2(X,\lambda)$

Suppose $S=\{f\in L^2(X,\lambda): f=\alpha_1( \chi_{A_1}-\chi_{X\setminus A_1})+\sum_{i=2}^N \alpha_ i\chi_{A_i}$ where $A_i, A_j$ are disjoint and $a_1>\max_{2\leq i\leq N} \alpha_i \}$. Is ...
0
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2answers
17 views

If $0<r < s$, then $L^s(\mu)\subseteq L^r(\mu)$. Under what conditions do these two spaces contain the same functions?

Suppose $f$ is a complex measurable function on $X$, $\mu$ is a positive measurable on $X$ and $||f||_p=(\int_X|f|^p d\mu)^{\frac{1}{p}}$ for $0<p<\infty$ If $\mu(X)=1$. $(i)$ $||f||_r\leq ...
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3answers
20 views

Find a sequence of measurable functions defined on a measurable set $E$ that converges everywhere on $E$, but not almost uniformly on $E$.

Find a sequence of measurable functions defined on a measurable set $E$ such that the sequence converges everywhere on $E$, but the sequence does not converge almost uniformly on $E$. I'm having ...
0
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1answer
20 views

$f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$.

If $f_n \in L^p\cap L^p{'}$ such that $p\neq p'$ and $f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$. a suggestion please.
1
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1answer
19 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
3
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1answer
45 views

Measure of the set $\{(x+f(x), x-f(x)):x\in \mathbb R\}$ in $\mathbb R^2$ is 0

Let $f:\mathbf R\to \mathbf R$ be a Lebesgue measurable function. Then the set $S=\{(x+f(x), x-f(x)):x\in \mathbf R\}$ is Lebesgue measurable in $\mathbf R^2$ and its measure is $0$. I am ...
1
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1answer
41 views

This one wierd trick integrates fractals. But does it deliver the correct results?

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method seems to vastly simplify the process (So even a comparative layman like me can do it). ...
1
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1answer
25 views

The Essential Supremum as a Limit

Let $(X, \mathcal F, \mu)$ be a finite measure space and let $f\in L^\infty(X, \mu)$. Define $\alpha_n=\int_X |f|^n\ d\mu$. Then $$\lim_{n\to \infty}\frac{\alpha_{n+1}}{\alpha_n}=\|f\|_\infty$$ ...
1
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1answer
37 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 ...
1
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1answer
18 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 ...
0
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0answers
24 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 ...
3
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1answer
28 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 ...
2
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0answers
30 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 $\mathcal{M}$ ...
0
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1answer
13 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
14 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 ...
1
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2answers
65 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 ...
0
votes
1answer
18 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 ...
1
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1answer
27 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 $$ ...
0
votes
1answer
25 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 ...
3
votes
1answer
27 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
34 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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0answers
16 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
25 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
18 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?
4
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1answer
30 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 ...
0
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0answers
27 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 ...
0
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1answer
43 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 ...
1
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1answer
23 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
34 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 ...
0
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1answer
31 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 ...
2
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0answers
39 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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0answers
13 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 ...
2
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2answers
33 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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0answers
12 views

Definition of an Algebra - Measure Theory

So an algebra of a fixed set $X$ is a collection of subsets of $X$ such that it is closed under complementation and unions of sets. So is the difference between an algebra and a $\sigma$-algebra ...
0
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1answer
21 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 ...
3
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0answers
39 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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0answers
16 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
2
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2answers
67 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 ...
1
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3answers
65 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

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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0answers
8 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
27 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 $\mathbb{R}^n$. ...
0
votes
0answers
13 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 ...
1
vote
1answer
37 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 ...
0
votes
1answer
41 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 ...