0
votes
0answers
43 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
0
votes
1answer
22 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
1
vote
0answers
17 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
2
votes
1answer
23 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
1
vote
0answers
31 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
1
vote
1answer
35 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
3
votes
3answers
57 views

Prove that every Lebesgue measurable function is equal almost everywhere to a Borel measurable function

Suppose $(\mathbb{R},\Sigma(m),m)$ is our measure space, where $m$ is Lebesgue measure. Also, suppose $f : \mathbb{R} \to [-\infty, \infty]$ is a Lebesgue measurable function. The problem: Prove ...
0
votes
1answer
32 views

Every Lebesgue measurable set contains a closed subset such that the set difference has small measure

M is the lebesgue measurable sets on $\mathbb{R}$. I have this exercise: Suppose that $E \in M$. Show that for each $\epsilon > 0$, there is a closed set F, with $F \subset E$ and $\lambda(E ...
0
votes
0answers
34 views

Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains an open interval [duplicate]

Suppose that $E$ and $F$ are Lebesgue measurable sets of $\mathbb{R}$, and their Lebesgue measures $m(E) > 0, m(F) > 0.$ Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains a nonempty ...
4
votes
1answer
40 views

$\int f = \lim\int f$ but $\int_{E}f\neq\lim\int_{E} f_{n}$

This is exercise 2.13 in Folland's Real Analysis textbook Let $(X, \mathcal{M})$ be a measurable space. Suppose $\{f_{n}\}\subset L^{+}$, $f_{n}\to f$ pointwise, and $\int f=\lim\int ...
0
votes
0answers
9 views

Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$

Let $A \subset \mathbb R^n$, prove that there is $H$: $A \subset H$, with $H$ a $G_{\delta}$ set such that $|A|_e=|H|$. The definition of $|A|_e$ is $|A|_e=\inf\{m(U): A \subset U\}$ where the ...
1
vote
0answers
58 views
+50

Is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
1
vote
2answers
32 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
2
votes
1answer
45 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
0
votes
1answer
20 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
1
vote
1answer
18 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
0
votes
0answers
27 views

Borel image of the projection [on hold]

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps Borel sets to Borel sets?
5
votes
1answer
131 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
votes
1answer
36 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
4
votes
2answers
221 views

Is this set measurable?

Let $E$ be a subset of $\mathbb{R}$. Assume that $\forall x\in E, x$ is a limit point of $E\setminus\{x\}$. Then, is $E$ Lebesgue-measurable? For example, any perfect subset, open subset or ...
0
votes
1answer
36 views

Proof that $X$ countable, $\mathcal M$ algebra on $X$ implies $\mathcal M$ a $\sigma$-algebra

In my measure theory class, I believe the professor made the claim that if $X$ was a countable or finite set and $\mathcal M$ was an algebra on $X$, then $ \mathcal M$ was a $\sigma$-algebra. I am ...
1
vote
2answers
58 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
3
votes
1answer
56 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
1
vote
0answers
20 views

Absolute continuity for non-measures?

Let $B$ be the collection of Borel subsets of $R^2$. A measure on $B$ is said to be absolutely continuous with respect to area if any subset with area 0 has measure 0. Is there a natural ...
0
votes
1answer
15 views

Absolute continuity of two-dimensional measures

Absolute continuity has two different meanings: one for functions and one for measures. The Wikipedia page explains the relation between the two notions in the following way: A finite measure μ ...
-4
votes
0answers
27 views

Determine the Lebesgue measure of the set of numbers without a fixed digit in its decimal expansion [closed]

Let $A \subset [0,1]$ be the set of all numbers without 9 over its decimal expansion. Determine the Lebesgue measure of $A$.
1
vote
0answers
113 views
+50

Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1} = \{x \in \mathbb{R}^d:|x|=1\}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every rotation $r$ of $\mathbb{R}^d$ and ...
2
votes
1answer
37 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
2
votes
0answers
36 views

Characterisation of absolutely continuous measure on the real line

Let $\lambda, \nu$ be two Radon measures on $\Bbb R$ such that $\lambda(\Bbb R)< \infty$. Show that the following are equivalent: $\lambda \ll \nu$; $\forall \epsilon>0$ there exists ...
4
votes
1answer
67 views

Characterization of measurability by closed sets.

If $E \subseteq \Bbb R$ is measurable, then for all $\epsilon > 0$, exists $F \subseteq \Bbb R$ closed such that $F \subseteq E$ and ${\frak m}^*(E \setminus F) < \epsilon$. I have already ...
2
votes
1answer
45 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
2
votes
2answers
44 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
0
votes
1answer
45 views

sigmal algebra and measure [closed]

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
0
votes
1answer
18 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
0
votes
1answer
52 views

Question about statement of Fubini's theorem

This is a question on the statement of Fubini's theorem for measurable sets. The theorem looks like this: Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda = \mu \times \nu)$ be a complete ...
3
votes
1answer
36 views

If $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, does that imply $(X, \Sigma, \mu)$ is $\sigma$-finite?

I'm having trouble proving or disproving the statement: If the product space $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, then so is $(X, \Sigma, \mu)$. I ...
0
votes
0answers
33 views

Prove the following is a generated algebra

Let $ \mathbb E $ = {$A_{1}$, $A_{2}$,.......$A_{n}$} $\subseteq$ $\mathbb P (X)$ given, if we define $A ^a$ = \begin{cases} A & \text{if }a=0,\\ A^c & \text{if }a=1 \end{cases} for each $ ...
2
votes
1answer
65 views

What exactly is a product measure?

If we have $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ a complete measure space with underlying complete spaces $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$, and $\lambda = \mu \times \nu$, what ...
0
votes
1answer
21 views

Prove a set which conatins one point from each class in circle of circumference 1 is nonmeasureable

(Kolmogrov,p268,problem 7) Let C be a circle of circumference 1 and let $\alpha\in\mathbb R\setminus\mathbb Q$. Let all points f C which can be obtained from each other by rotating C through an ...
2
votes
1answer
65 views

Inequality of intervals lengths.

Let $(I_j)_{j = 1}^n$ be a finite collection of intervals that covers the rationals in $[0, 1]$. Prove that $\sum_{j = 1}^n \ell (I_j) \geq 1$. (Here, $\ell (I)$ denotes the length of the interval ...
2
votes
1answer
31 views

If $f \in L^{1}(d\mu)$, is it true that $\int \limits_{X} f\chi_{\{ f \neq 0 \} } \,d\mu = \int \limits_{X}f \,d(\chi_{\{ f\neq 0 \} }\,d\mu)$?

Ok, so we have $f \in L^{1}(d\mu)$, with $(X, \Sigma, \mu)$ a complete measure space. If we assume $f$ is nonnegative, we can define a measure $\rho(E) = \int \limits_{E} f \,d\mu$ for $E \in ...
3
votes
1answer
40 views

If $f \in L^{1}(d\mu)$ is nonnegative, can we conclude $\mu( \{ x \mid f(x) \neq 0 \} ) < \infty$?

I am trying to prove a statement, and I need the fact that: If $f \in L^{1}(d\mu)$ is a nonnegative function, then this implies $\mu( \{x \mid f(x) \neq 0 \} ) < \infty$. But I don't know ...
3
votes
1answer
49 views

Measure Theory Inequality

I was having trouble showing the following inequality: Prove that if $A \subset I = [0,1]$ has measure $u(A) < 1$ and $\epsilon > 0$, then there is an interval $[a,b] \subset I$ such that $u(A ...
1
vote
1answer
31 views

Lebesgue measurable functions and the absolute value of them

Let $f$ is a measurable function. If $f$ is Lebesgue integrable, is the absolute value of $f$ Lebesgue integrable?
0
votes
1answer
31 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
1
vote
1answer
29 views

If $E, F \subset [0, 1]$, $m(E), m(F) > 0$, and $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$, show $m(F\cap E_n) > 0$ for sufficiently large $n$

Suppose $E \subset [0, 1]$ has positive Lebesgue measure and let $E_n = \{x \in [0, 1] : nx \bmod 1 \in E\}$. If $F \subset [0, 1]$ has positive Lebesgue measure, show that so does $F \cap E_n$ for ...
2
votes
0answers
34 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
6
votes
1answer
131 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
2
votes
2answers
55 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
1answer
45 views

Where is “countability” used in this proposition about product $\sigma$-algebra?

The following is a proposition about product sigma algebra from Folland's Real Analysis: Proposition. If $A$ is countable, then $\otimes_{\alpha\in A}M_{\alpha}$ is is the $\sigma$-algebra ...