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

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Show that $\int f d(\lambda + w \mu) =\int f d \lambda + \int fw d \mu $, where $w $ is a bounded function.

I don't understand the following, from Rudins real and complex analysis. Assume $\lambda $ is a positive bounded measure, and associate to the $\sigma-$finite measure $\mu $ a function $0< w(x) ...
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0answers
42 views

$f$ absolutely convergent on $\mathbb{N}$ iff integrable

I am looking for some hints on the following Let $X$ be the set of all positive integers, $\Sigma$ the class of all subsets of $X$, and $\mu(E)$ (for any $E\in \Sigma)$ the number of points in $E$. ...
2
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1answer
37 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
2
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1answer
47 views

continuous-time version of fatou's lemma

I have just read a textbook on stochastic processes that implicitly uses the fact that \begin{equation} \int \liminf_{t \to \infty} f_t \leq \liminf_{t \to \infty} \int f_t, \end{equation} for ...
2
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0answers
28 views

Passing to the limit in equation with null sets (related to Bochner space)

Let $S \subset L^2(0,T;V)$ be a subset with the embedding dense, where $V$ is separable Hilbert space. Let $f:[0,T]\times V \to \mathbb{R}$ be continuous in the second argument. Suppose that for all ...
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1answer
27 views

Simple function sequence for measurable functions with range $[0,1]$.

Let $[0,1]$ have the usual topology. Consider $f: X \rightarrow [0,1]$ such that $f$ is measurable. Fix $n$. I want to show that there is a measurable simple function $\phi_{n}$ such that ...
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1answer
26 views

Supremum of measurable functions

If $\{f_n\}$ is a sequence of measurable functions on the same measurable set then: i) $\sup_{1\le i\le n} f_i$ is measurable for each $n$. ii) $\sup f_n$ is measurable. I don't quite follow the ...
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2answers
52 views

Sequence of Measurable Functions (Unsigned and Complex-Valued)

I am having several difficulties in solving the following problem about measurable functions: Let $(X,\mathcal{B})$ be a measurable space. If $f_n : X \to [0,+\infty]$ are a sequence of measurable ...
3
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1answer
55 views

Borel measurable functions in measure theory

Suppose that $f$ is a function on $\mathbb{R}\times \mathbb{R^k}$ such that $f(x,\cdot)$ is Borel measurable for each $x\in \mathbb{R}$ and $f(\cdot,y)$ is continuous for each $y\in \mathbb{R^k}$. For ...
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2answers
27 views

What is meant with a $\sigma $-finite measure $\mu $ can be replaced by finite measure in the sense that $d \bar {\mu}=w d \mu $?

I don't understand the following. Before proving the Radon-Nikodym theorm, Rudin proves a lemma that say that $\mu $ is a $\sigma $-finite measure on a $\sigma $-algebra, then there is a function $w ...
3
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1answer
37 views

Convergence in equivalent probability measure

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A_n$ be a sequence of events such that $P(A_n)$ converge to 0. If $Q$ is an equivalent probability measure of $P$, does it mean that $Q(A_n)$ ...
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1answer
121 views

Infinite product of probability measures is a premeasure

This is an exercise from Real Analysis by Stein (Chapter 6, Exercise 15). Given infinitely many measure spaces $X_i$, each of which has measure 1, one can define an algebra on the product space ...
2
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0answers
44 views

For a measure $\mu $ *does it exists* a sequence $\{E _n \} $ of sets such that $\mu(E _n)<1/2^n $ for every $n $?

If $\mu $ is a measure defined on a measure space $(X, M, \mu ) $, can I claim that there exists a sequence $\{E _n \} $ of set in $M $ such that for every $n $, $\mu ( E _n) <1/2^n $? If so, what ...
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0answers
14 views

Measurability of $u:[0,T] \to X$ where $u$ is in Bochner space

Let $f \in L^p(0,T;X)$ where $X$ is a separable Banach space. So $f$ is a Bochner function and hence Bochner measurable, meaning that there is a sequence of measurable countably-valued functions that ...
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1answer
31 views

Problem in the proof of the dimension of the Cantor set

From the proof of the Hausdorff dimension of the middle third Cantor set. I cannot understand the last sentence in this proof. I cannot see how we have counted $2^j \leq \sum_i 2^j3^s|U_i|^s$
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1answer
44 views

Monotone functions and Borel sets

I'm studying measure theory and two question came to my mind: If $f:\mathbb{R}\to\mathbb{R}$ is monotone and $B\subseteq\mathbb{R}$ is borel, is the image $f(B)$ borel? If ...
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1answer
61 views

Writing $\{x; (fg)(x)>a\}$ as a countable union of measurable sets for measurable functions $f,g$

I was trying to prove that the product of two measurable functions on the extended real line is measurable and I came across this statement: \begin{equation*} \{x : (fg)(x) > a\} = \left(\bigcup_{r ...
4
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3answers
164 views

Help with conditional expectation on the circle

Let $p >1$ a integer, $X = \mathbb{R} / \mathbb{Z}$ and $\mu\colon \mathcal{B}\to [0,1]$ a probability measure on the Borel subsets of $X$ which is $T \colon X \ni x \to (px \text{ mod }1)$ ...
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0answers
34 views

Measures: What is the relation between almost everywhere and the empty set?

For a positive measure $\mu$ on a $\sigma$-algebra $\mathfrak M$, we have that $$\mu(\emptyset)=0$$If $P$ is a property which a point may or may not have. The statement "P holds almost everywhere on ...
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1answer
45 views

Almost sure convergence for measurability

So it is pretty obvious that the limit of measurable random variables is also measurable if they converge...but I'm not sure it's true of almost sure convergence. Suppose there is a probability space ...
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1answer
31 views

find a goed function for dominated convergence theorom and solve the integral [closed]

Let a>=0 and define the functions $f_n$ : [0,1)$\rightarrow$R by $f_n(x)$=($n^{2}xe^{{-n^2}{x^2}}$)/($1+x^{2}$) determine the limit $\lim_{n\to\infty}$$\int^\infty_a $$f_n(x)$dx for the cases a ...
2
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0answers
26 views

Differents definitions of integral

In some books (like Rudin's or Cohn's) defined the integral of a function $f\geq 0$ by $$\int f \text{ } d\mu =\sup\{\int \phi \text{ } d\mu;\phi\leq f \wedge\phi \text{ is a simple function} \}$$ ...
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1answer
37 views

Assumptions of the MCT

Monotone Convergence Theorem: Let ($f_n$) be a sequence in $\Sigma^+$ (i.e. measurable and nonnegative), such that $f_{n+1}\geq f_n$ almost everywhere for each $n$. Let $f=\limsup_n f_n$. Then $\mu ...
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2answers
37 views

$\lim_{y\to\infty}\int_{0}^{\infty} (y\cos^2(x/y))/(y+x^4) \, dx$

How do I calculate the limit $$ \lim_{y\to\infty}\int_{0}^{\infty} \frac{y\cos^2(x/y)}{y+x^4} \, dx? $$ It's about measure theory. I though about Fatou's lemma, but I couldn't solve it.
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1answer
25 views

Problem with Hausdorff dimension

Let $\varepsilon>0,s\geq0$ and $C\subseteq \mathbb{R}^d$ be randomly given. Now define: $$ \mathcal{H}^s_\varepsilon(C)= \inf\biggl\{\,\sum^\infty_{n=1}(\rho (A_n))^s\biggm| C\subseteq ...
0
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1answer
21 views

Continuity of an integral operator

I'm stuck with this exercise: Let $A \subset \mathbb{R}$ be a measurable set. For each $f \in L^1(\mathbb{R})$ and $y \in \mathbb{R}$, let: $T(f, y) = \int_{A}{f(x-y)\mathrm{d}x}$. I have to show ...
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1answer
23 views

Equivalence of definition of measure concentrated on a set $A $

Let $\lambda $ be a positive or complex measure. We say that $\lambda $ is concentrated on $A $ if for some set $A \in \mathcal{B } $ we have that $\lambda (E) =\lambda (A \cap E ) $ for every $E \in ...
3
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1answer
76 views

Linear combinations of indicator functions of measurable rectangles with sides of finite measure.

The following is a question on measure theory whose answer would help me greatly in my research. Let $ (X,\mathcal{F},\mu) $ and $ (Y,\mathcal{G},\nu) $ be $ \sigma $-finite measure spaces. Let $ ...
0
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1answer
53 views

Why is this set needed in this proof?

The theorem states that if $\{ f_n\}$ is a sequence of complex measurable functions defined almost everywhere on a set $X$ such that $$\sum_{n=1}^{\infty}\int_{X}\vert ...
2
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1answer
60 views

extension to measure, why do they not need to assume countable additivity

Here they present some necessary conditions in order to be able to extend to a measure: But lets say you have a disjoint sequence $\{A_1,A_2,\ldots\}$ whose union happens to be in the set you want ...
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2answers
44 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
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2answers
34 views

Outer Measure Subadditivity

I'm having trouble constructing a sequence $\{E_n\}$ of disjoint subsets of $\mathbb{R}$ such that $$m^{*}\left(\bigcup_{i}E_i\right) < \sum_{i}m^{*}(E_i).$$ What's a way to gain some intuition ...
3
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0answers
62 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
2
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1answer
51 views

Lebesgue measure and compact set

Let $A \subset\mathbb R^d$ be a Lebesgue measurable set and bounded. Show that $$m(A) = \sup\{m(K) : K \subset A\text{ and }K\text{ compact} \} \in [0,\infty].$$ answer : I know that for each ...
3
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0answers
57 views

Can $\|f\|_p\to\infty$ arbitrarily slowly? (Looking for hints.)

Given $f$ is Lebesgue measurable on $(0,1)$ and not essentially bounded, is it true that to every positive function $\Phi$ on $(0,\infty)$ such that $\Phi(p)\to\infty$ as $p\to\infty$ one can find an ...
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1answer
36 views

Hausdorff measure of the middle third Cantor set and Compactness

In the proof of the Hausdorff dimension of the middle third cantor set I cannot understand why we need the following underlined statement. I cannot understand why we need only consider closed ...
3
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3answers
48 views

Does a set of positive outer measure contain a measurable subset of positive measure?

Is it true or false that whenever $E \subseteq \mathbb{R}$ is such that $m^*(E) > 0$, where $$ m^*(E) = \inf\left\{\sum_{n=1}^\infty|b_n - a_n|\, :\mid\, E \subseteq \bigcup_{n = ...
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1answer
54 views

Two questions on Banach function spaces

I have recently started studying Banach function spaces over $\sigma$-finite measure spaces. By a Banach function space I mean: Let $\left(R, \mu \right)$ be a $\sigma$-finite measure space and let ...
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23 views

proof in Holders inequality,(equality) [duplicate]

I have this proof in my book: I would like to prove what I underlined in red. but I get stuck. I guess in order to get equality we only need the opposite inequality. However I still don't ...
2
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0answers
42 views

Relation of stopped sigma-Algebra on cadlag sample space to arbitrary sample space

Let $X_t : \Omega \to E$ be a cadlag process with Polish state space $E$, $T$ a stopping time w.r.t. the canonical filtration $\mathcal{F}_t$ of $X$ and $X^T_t$ the stopped process. Then it should ...
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1answer
35 views

Determine $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$, and is $u$ $\mathcal B(\mathbb{R})/\mathcal B(\mathbb{R})$-measurable?

Let $u:\mathbb{R}\to\mathbb{R}$ be given by $u(x)=\left \lfloor x \right \rfloor$. Determine the set $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$. Show that $u$ is $\mathcal ...
0
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1answer
16 views

Meaning of “a distribution does not live on any proper subinterval of $(-\infty,\infty)$”.

While I am reading a probability note related to Radon-Nikodym derivative, I found the sentence I could not understand easily: "Assume that the distribution $\mu$ does not live on any proper ...
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1answer
37 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
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1answer
55 views

Lesbegue Outer Measure

Consider the unit interval $I=[0,1]$ and let $\mathcal{M}$ be the $\sigma$-algebra of all Lebesgue measurable subsets of $I$. Denote by $m_*$ the Lebesgue outer measure on $\mathcal{M}$. Suppose that ...
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0answers
70 views

Real Analysis texts: Royden versus Stein & Shakarchi. Which is better? (and other suggestions welcome)

I am taking an introductory "graduate" analysis class and am comparing Analysis books that cover measure theory. I have had an "advanced calculus" class that covered the standard topics. I am having ...
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2answers
42 views

Decomposing Countable Union of Measurable Sets

Why can every set $E$ in the real numbers with $\mu^{*}(E)=\infty$ be realized as the disjoint union of countably many measurable sets, each of which has finite outer measure? I'm trying to see this ...
3
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1answer
35 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
5
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2answers
96 views

Which integral is larger?

The question: Given $f$ to be a positive, measurable function on $[0,1]$, which is larger, $\displaystyle\int_0^1 f(x)\log f(x)\,dx$ or $\displaystyle\left(\int_0^1f(s)\,ds\right)\left(\int_0^1\log ...
0
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2answers
83 views

Complex Measures: Total Variation Measure Decomposition

Let $\mu:\Sigma\to\mathbb{R}$ be a complex measure. Does the total variation measure admit a decomposition: $$\mu(E)=|\mu|(E\cap A)-|\mu|(E\cap B)+i|\mu|(E\cap C)-i|\mu|(E\cap D)$$ with $A,B,C,D$ ...
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0answers
30 views

Neighborhood near $0$ for a sequence of non-negative measurable functions can be made arbitrarily small in measure.

I am doing some practice exercises from Avner's FoMA (2.4.1 to be precise). And encountered the following: Let $(X,\Sigma, \mu)$ be a finite measure space. Let $\{f_n\}$ be a sequence of a.e. ...