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

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0
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1answer
33 views

Existence of a function from $[0,1]$ to an arbitrary measurable set

For any measurable $E\subset \mathbb{R}$ with measure 1, does there exist a continuous function $$T:[0,1]\to E$$ Such that $\mu\circ T=\mu$, where $\mu$ is Lebesgue measure on the real line?
1
vote
2answers
69 views

proving inequality involving lebesgue measurable function on [0,1]

Just by looking at the inequality, I can tell that if $\displaystyle \int_0^1 f(x)dx \geq 1$ then $\displaystyle \int_0^1 \frac{dx}{f(x)} \leq 1$. The same thing vice versa. How do I prove that, ...
1
vote
1answer
35 views

Does this measure make it such that $x^3 \ge 1$ almost everywhere?

$x^3 \ge 1$ almost everywhere $\Longleftrightarrow \mu(\{x : x^3 < 1\}) = 0$ $\Longleftrightarrow \mu((-\infty, 1)) = 0$ Let $$\tau(E) = \int_Efdm$$ with $f = \chi_{(2, 3)}$ So $f$ is a ...
3
votes
0answers
151 views

Applying Dominated Convergence Theorem to solve $\lim_{n \to \infty}\int_0^n(1-\frac{x}{n})^ne^{-x}dx$

$$\lim_{n \to \infty}\int_0^n(1-\frac{x}{n})^ne^{-x}dx$$ I am going to use the Dominated Convergence Theorem to solve this. $$\lim_{n \to \infty}\int_0^{\infty}\chi_{[0, ...
0
votes
0answers
41 views

Question regarding the Monotone Class Theorem

The statement of the Monotone Class Theorem is as follows: Given an algebra $\mathcal{A}$ denote by $\mathcal{M}\left(\mathcal{A}\right)$ the monotone-class generated by $\mathcal{A}$. Then ...
1
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2answers
165 views

Application of the Dominated Convergence Theorem

$$lim_{n \to \infty} \int_0^1(1 - e^{\frac{-x^2}{n}})x^{-1/2}dx$$ I want to use the Dominated Convergence Theorem to solve this. Let $f_n = (1 - e^{\frac{-x^2}{n}})x^{-1/2}$. Step 1: Determining ...
1
vote
3answers
129 views

Why is this set a Borel set on $R^2$?

Consider a measurable space $([0,1]\times [0,1], \mathcal{B}([0,1]) \times \mathcal{B}([0,1]))$, and a subset $A:=\{(x,y):x=y\}$ (the diagonal). According to the text book, $A \in \mathcal{B}([0,1]) ...
0
votes
1answer
71 views

Show that if $\frac{1}{\mu\left(E\right)}\cdot\int_{E}fd\mu\in C$ for all $E$ with $\mu\left(E\right)>0$ then $f\left(x\right)\in C$ almost surely

I'm trying to prove the following claim but I really don't know where to start: Let $\left(X,\mathcal{F},\mu\right)$ be a finite measure space and let $f:\left(X,\mathcal{F}\right)\to\mathbb{R}$ ...
1
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0answers
46 views

Monotonicity of a probability sequence

Let $\mathbb{P}$ be a probability measure on $\mathbb{R}^n$, so $\mathbb{P}( \mathbb{R}^n ) = 1$. Consider a mapping $f: \mathcal{P}( \mathbb{R}^n ) \rightarrow \{ 0, 1, ..., F \}$, where ...
3
votes
1answer
68 views

What is measure of the following set?

Suppose $ D= \lbrace (x,y) \in [0,1]\times [0,1]: x-y \in \mathbb{Q} \rbrace $. Is the measure of $D$ zero? Thanks.
1
vote
1answer
70 views

Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere. [duplicate]

Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere. I was thinking that maybe the finite sum could be compared to a simple function ...
1
vote
2answers
59 views

$l^q \subset l^p$ for $q\leq p$ with counting measure

Suppose $\Sigma = \mathcal{P}(\Omega) < \infty$ and $\mu$ is the counting measure. I'm seeking to show that $l^q \subset l^p$ for $1 \leq q\leq p \leq \infty$. The main obstacle to a proof is ...
0
votes
2answers
131 views

Associativity of Product Sigma Algebra

\begin{align} Let (X,\mathscr{A}, \mu), (Y,\mathscr{B},\nu) ,\space and\space (Z,\mathscr{C},\theta)\space be\space measure\space spaces.\space Show\space that\space ...
-2
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1answer
49 views

Find a set similar to cantor set [closed]

I need to prove that there is a compact subset of [0,1] with an empty Interior with measure 1, other than Cantor set?
3
votes
0answers
30 views

Question about inner measure

I need to prove that for every $A,B\in\mathbb{R}$: $m_*(A)+m_*(B)\geq m_*(A\cup B) +m_*(A\cap B)$ (When $m_*$ is the inner measure). I have no idea how to do this...
4
votes
2answers
131 views

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution.

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution. To differentiate $fg$, we may apply Leibniz's rule ( ...
6
votes
1answer
54 views

Can You Construct a Syndetic Set with an Undefined Density?

Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i ...
1
vote
1answer
101 views

Contour integral in complex plane as a Lebesgue integral

In an occasion, I'd like to use Fubini's Theorem to swap the order of integration of a countour integral with an integral with respect to a measure (show that $\int_{\gamma} \int_{\Omega} f(x,z) ...
0
votes
1answer
61 views

A Question on Lebesgue Decomposition

How can I show that the Lebesgue measure has no Lebesgue decomposition with respect to the counting measure? I know how to show that reverse is true, but just can't figure out how to show this ...
9
votes
1answer
113 views

Evaluating $\sum_{n=0}^{\infty}\ \int_{\pi/4}^{\pi/3}\sin^{n}x (1-\sin x)^2 dx$ using a convergence theorem

$$\sum_{n=0}^{\infty}\ \int_{\pi/4}^{\pi/3}\sin^{n}x (1-\sin x)^2 dx$$ Let $g_n = \sin^{n}x (1-\sin x)^2$ $g_n$ is a sequence of measurable functions and $g_n \ge 0$ so applying the Beppo Levi ...
1
vote
0answers
34 views

Is this a measure on $\mathbb{R}$ that makes $x^3 \le 8$ a.e. true

So I'm looking for a measure on $\mathbb{R}$ that makes $x^3 \le 8$ almost everywhere true. $\Longleftrightarrow$ I want a measure $\mu$ such that $\mu \{x: x^3 > 8\} = 0$. $\Longleftrightarrow ...
1
vote
0answers
42 views

Well - order on reals and not Lebesgue measurable sets

Let $(x_t )_{t<\omega_1} $ be a well ordering of the set of real numbers of the type $\omega_1 $ ( the first uncountable ordinal, we assume the continuum hypothesis ) and let $\leq $ be a natural ...
0
votes
0answers
41 views

I want to show (i) and (ii) are equivalent.

Show that the following are equivalent: (i) A Family $A_{i}$ of events is independent; (ii) The family $\sigma(\mathbb 1_{A_{i}})$ of $\sigma$-algebra is independent. thanks for help.
1
vote
1answer
31 views

Trouble with intuition of convergence in measure

I'm having trouble trying to "read" what exactly convergence in measure is meaning. Specifically, I'm working on showing $f_n = n^{-1/p}1_{[0,n]}$ converges in measure to $0$ on $(\mathbb{R}, ...
1
vote
1answer
75 views

Absolutely continuous, strictly positive measures

Let $\mu$ be a strictly positive $\sigma$-finite measure on $\mathbb{R}^{n}$ that is absolutely continuous with respect to the Lebesgue measure, $\lambda$. One way to think about the absolute ...
0
votes
1answer
26 views

How can I show the equality of integration for shifting simple functions over $\mathbb{R}$

Let $\phi(x) = \sum_{k=1}^n a_i\chi_{E_i}(x)$ be a simple function on $\mathbb{R}$ with finite support. I want to show that \begin{equation} \int_\mathbb{R} \phi(x) = \int_\mathbb{R} \phi(x+t). ...
2
votes
1answer
142 views

Lebesgue's Dominated Convergence Theorem questions

Assume that $f_n \to f$ almost everywhere, with $f_n$ integrable for all $n$ and $g$ is an integrable function such that $\lvert f_n \rvert \le g$. (A) Then $f$ is integrable and - $$\int f\,\mathrm ...
0
votes
1answer
49 views

0-1 Law: Applications?

The question is more open to a debate rather than a mathematical explanation: If $(A_n)_{n\in \mathbb{N}}$ is a sequence of $\sigma$-algebras. $\mathfrak{A}_n := \sigma(\bigcup_{m\ge n} A_m)$ is the ...
1
vote
1answer
54 views

Question on Fatou theorem

Can we replace inequality: $$\int_X f(x)\text{d}\mu \leq \underline\lim_{n\to\infty} \int_X f_n(x) \text{d}\mu$$ in Fatou's lemma replace by equality. If not, how can it be proved?
5
votes
1answer
257 views

Haar measure on the groups SO(n) and SO(n,m)

Would you please give me some information about Haar measure on special orthogonal group SO(n) and indefinite special orthogonal group SO(n,m)? Thank you so much!
3
votes
1answer
33 views

Show: $\mathcal{L}^p\subset\mathcal{L}^q$

Consider a measurable space $(\Omega,\mathcal{A},\mu)$. Let $\mu(\Omega)<\infty$ and $1\leq q\leq p$. Show that then $$ \mathcal{L}_{\mu}^p\subset\mathcal{L}_{\mu}^q. $$ Good ...
1
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0answers
63 views

Integral depending on parameter is continuously differentiable

Let $\phi\in\ L^1([0,1])$ and $f:\mathbb R\rightarrow\mathbb R$ defined by integral $f(t)=\int_{0}^{1}|\phi(x)-t|\ dx$, show that $m(\{\phi=t\})=0\ ,\forall t\ ,in\ an\ ...
1
vote
1answer
52 views

LDC Theorem and Hardy-Littlewood Function question

I'm trying to show that for any $r >0$ and $f \in L^1$ then $A_{f,r} := \frac{1}{m(B(x,r))} \int_{B(x,r)} f(y)dm(y)$ is also $L^1$. In fact we can show that $\|A_{f,r}\|_1 \leq \|f\|_1$ where ...
1
vote
2answers
97 views

Integration of non-negative measurable functions

Let $f$ be a non-negative measurable function whose domain is $X$. The the integral of $f$ is defined as - $$\int_X f \mathrm d\mu = \sup\left\{\int_X s \mathrm d\mu: 0 \le s \le f\right\}$$ where ...
1
vote
1answer
17 views

Show that there exitst $f \in L^{1}([0,1])$ such that $\int_{0}^{1}f(x)g(x)dx \nrightarrow 0$

Define $$ g_{n} = n\mathbb{I}_{[0,\frac{1}{n^3}]}(x)\;\; $$ where $\mathbb{I}$ is index function. (if $x \in E, \mathbb{I}_{E}(x) = 1$, otherwise 0) show that there exists $f \in L^{1}([0,1])$ such ...
2
votes
2answers
56 views

Borel Not Complete

I am a beginner in measure theory and I read that "Not every subset of a set of Borel measure $0$ is Borel measurable". Suppose Borel set is $B$. then is there is some $A$ in $B$ such that $m(A) = 0$ ...
2
votes
1answer
73 views

Lipschitz-continuity and measurability

Problem: Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be Lipschitz-continuous with constant $L$. Let $\lambda_n$ be the $n$-dimensional Lebesgue-measure. Show that there exists a constant $C\ge 0$, s.t. ...
1
vote
1answer
39 views

Measure theory - image measure question

If $f:E\to G$ is a measurable function between two measurable spaces and $\mu$ is a measure on $E$, I am considering the image measure $\nu = \mu \circ f^{-1}$. I want to show that $\nu(g) = \mu(g ...
0
votes
0answers
16 views

Elementary measure theory exercise about $\liminf_{n \to \infty} \mu (E_n)$ [duplicate]

I'm going through the Fremlin's book (vol 1) on my own, and frankly, I don't have any good idea where to begin with the following exercise: Assuming that ($X, \Sigma, \mu$) is an measure space and ...
1
vote
1answer
61 views

Lebesgue's theorem for subsequence

Let $(\Omega, \mathcal A, \mu)$ be a measurable space with a finite measure $\mu$. Let $(f_n)_{n\in\mathbb N}$ and $f$ be real measurable functions such that $|f_n|\leq g$ for all ...
0
votes
1answer
47 views

Convergence question in measure theory

I have a convergence question in measure theory that requires assistance: Let $1\leq p<\infty$. Suppose $f,\ f_n \in L^P$, and $f_n\to f$ in $L^P$. (i.e $(\int|f_n-f|^pd\mu)^{1\over p}\to 0$ as ...
1
vote
0answers
60 views

Exercise 3.23 from Real and Complex Analysis of Walter Rudin [duplicate]

I need help in the exercise cited in the title. Suposse $ \mu $ is a positive measure on $ X $, $ \mu(X)<\infty $, $ f \in L^\infty(\mu) $, $ \|f\|_\infty >0 $, and \begin{equation} \alpha_n = ...
2
votes
1answer
51 views

Lebesgue Domination Convergence and $L^1$ Convergence

I'm working on a problem in my notes about LDC implying $L^1$ convergence. We will let $m$ denote Lebesgue measure and $\| \cdot \|_1$ denote $L^1$ norm. Further for $f \in L^1(\mathbb{R^n})$ and ...
3
votes
1answer
50 views

Why is this function continuous in the strip and analytic in it's interior? (Rubel's proof of Holder's Inequality)

Rubel's proof of Holder's Inequality is done in the following way: Given the measure space $(\Omega, \mathcal{F}, \mu)$, functions $f \in L^p$, $g \in L^q$ and conjugate exponents $p,q$, define ...
5
votes
1answer
220 views

Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...
1
vote
1answer
59 views

Surely $\Rightarrow$ almost surely?

I have a question that maybe is silly, but important. In our reading, we had (concerning the Lebesgue integral): Let $f,g\geq 0$ be measurable. Then: $$ f\leq g\mbox{ a.s. ...
4
votes
1answer
65 views

Probability space proof

PROBLEM Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(E_n)$ be a sequence in the $\sigma$-algebra $\mathcal{F}$. $a)$ If the sequence $(E_n)$ is increasing (in the sence that ...
6
votes
1answer
79 views

Measurability of the pushforward operator on measures

Let $X$, $Y$ and $Y'$ be (standard) Borel spaces. We let $\mathcal B(X)$ be the Borel $\sigma$-algebra of $X$ and $\mathcal P(X)$ to be the space of all Borel probability distributions on $X$ endowed ...
3
votes
1answer
57 views

Does convergence of $\lim_{n \rightarrow \infty} \int h d\mu_n$ for all continuous, bounded $h$ imply weak convergence of $(\mu_n)$?

Let $(\mu_n)$ be a sequence of positive finite Borel measures on $\mathbb{R}$. Suppose $\lim_{n \rightarrow \infty} \int h d\mu_n$ converges for every bounded, continuous function $h$ on $\mathbb{R}$. ...
2
votes
1answer
193 views

Does weak convergence with uniformly bounded densities imply absolute continuity of the limit?

Suppose $(f_n)$ is a sequence of probability density functions on $\mathbb{R}$ such that \begin{align*} f_n &\leq M \text{ for all }n \\ f_n(x) &= 0 \text{ for all } |x| > 1 \end{align*} ...