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

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2
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0answers
39 views

Particular $L^p$ space

I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and ...
2
votes
0answers
23 views

Integral of addition of measurable functions

Let $(X, \mathcal{M},\mu)$ be a measure space. Let $f,g: X \to \mathbb{R}$ (extended real line) be measurable functions. Prove that if both $\int f^{+} d\mu$ and $\int g^{+} d\mu$ are finite, or ...
1
vote
0answers
29 views

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
4
votes
2answers
40 views

Prove the following inequality: $\int_{(a,b)}f\ d\lambda\cdot\int_{(a,b)}\frac{1}{f}d\lambda≥(b-a)^2$

Assignment: Let $-\infty < a < b < \infty$ and $f: (a,b) \rightarrow (0,\infty)$ be measurable, such that $f$ and $\frac{1}{f}$ are Lebesgue integrable. Prove the following inequality: ...
0
votes
1answer
30 views

Show that $L^p(\mathbb{R}^d)$ spaces are not comparable one another.

I have to show that, in $\mathbb{R}^d$ with Lebesgue measure, the $L^p$ spaces are not comparable one another. More precisely, I want to show that given $p$ and $q$ such that $1\leq p<q\leq\infty$, ...
2
votes
0answers
19 views

An application of Strassen's theorem

Recently I handed in a problem set containing the following question, but neither myself nor my classmates managed to find a satisfying solution. We were quite certain that a fruitful approach was to ...
1
vote
1answer
42 views

Show that it is Lipschitz

Let $E \subset \mathbb{R}^d$ be Lebesgue measurable und let $\phi (t)=m \left ( \Pi_{i=1}^{d} (-\infty , t_i ) \cap E \right )$. I have to show that $\phi $ is Lipschitz. Could you give me some ...
0
votes
1answer
30 views

Spectral theorem question

I am trying to understand how to develop the spectral measure of a bounded self-adjoint operator on a Hilbert space. For every continuous function on its spectrum, $f: C(\sigma(A)) \to \mathbb{C}$, ...
1
vote
1answer
24 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
1
vote
1answer
20 views

Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT. However, I couldn't find another example when the measure is finite. Could ...
1
vote
1answer
29 views

The measure of the boundary being zero implies the set is measurable.

Assuming our set, $E\subset\mathbb{R}^2$ such that $m(\partial E)=0$ (where $m$ is Lebesgue measure), why does this imply that $E$ is Lebesgue measurable?
2
votes
1answer
31 views

Proving that a specific function is $L^1$.

From Rudin's Real & Complex analysis text: If $f$ is continuous on $\mathbb{R}^1$ with $f(x)>0$ for $x\in(0,1)$ and zero otherwise, if we define $h_c(x)=\sup\{n^cf(nx):n=1,2,3,\ldots\}$, how ...
0
votes
0answers
18 views

The usage of Radon-Nikodym theorem

A question from the qualifying exam I am studying: Let $(X,\mathscr{F},\mu)$ be a measure space, and suppose $\mathscr{E}$ is a sub $\sigma$-algebra of $\mathscr{F}$, that is, $\mathscr{E}$ is itself ...
1
vote
1answer
30 views

Does it stand that $\lim \inf |f_n|^p=|f|^p$ for this reason?

We have that $f_n, f \in L^p, 1 \leq p < +\infty$, $f_n \rightarrow f $ almost everywhere and $||f_n||_p \rightarrow ||f||_p$ . Do we have that $\lim \inf |f_n|^p=|f|^p$ because of the following?? ...
0
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0answers
38 views

Extension of Radon Nikodym Theorem(Folland Real Analysis Chapter 3 Problem 14)

If $\nu$ is an arbitrary signed measure and $\mu$ is a $\sigma -$ finite measure on $(X,M)$ such that $\nu<<\mu$, there exists an extended $\mu-$ integrable function $f:X \rightarrow ...
2
votes
1answer
34 views

interchange the summation with the integration.

We have $$\int_0^\infty e^u h(u) \ du = \int_0^\infty \sum_{k=0}^\infty \frac{u^k}{k!} h(u) \ du = \sum_{k=0}^\infty \frac{1}{k!} \int_0^\infty u^kh(u) \ du$$ Under what conditions we can ...
2
votes
0answers
34 views

Generalization of the Riesz-Markov theorem

So, my professor mentioned a version of the Riesz-Markov theorem for some kind of general spaces, that yields a maximum and a minimum measure rather than a unique measure (or something along those ...
2
votes
1answer
32 views

Show that the $\sigma$-algebras are independent

Suppose $(\mathfrak{B}_n)_n$ are independent with respect to a finite measure $\mu$. Show that for any $N$, the $\sigma$-algebras ...
1
vote
1answer
26 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
2
votes
1answer
47 views

Cardinality of a set of positive Lebesgue measure

I have pretty no knowledge in set theory, so likely the question has a trivial answer. All countable subsets of $[0,1]$ have Lebesgue measure of zero, thus all sets of positive Lebesgue measure are ...
2
votes
1answer
60 views

given $\epsilon >0$ , there is an $A\in \mathfrak{a}$ with $\overline \mu (A-E)+\overline \mu (E-A)<\epsilon$

Let $\mu$ be a measure on an algebra $\mathfrak{a}$ and $\overline \mu $ the extension of it given by the Caratheodory process. Let $E$ be measurable with respect to $\overline \mu $ and $\overline ...
2
votes
1answer
33 views

I'm confusing with inequality of p-norm

As far as I know, for $p \ge 1$, $||X||_p \equiv (E|X|^p)^{1/p}$ becomes a norm in probability space. If this is right, those two inequalities on each link seem to contradict with each other. ...
2
votes
1answer
37 views

Benefits of alternative formulation of the integral (general measure)

Let $(X,\mathcal{M},\mu)$ be a measure space and $f:X\to \mathbb{R}^+$ a measurable map. Set $$F(t) = \mu (\{x\in X \mid f(x)>t\})$$ and define $$\int_X f\,d\mu := \int_0^\infty F(t)\,dt,$$ ...
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0answers
33 views

Monotone convergence theorem allows the limit to be infinity

Monotone convergence theorem(MCT) doesn't impose any restriction on the limit. For example, if $\{X_n\}$ satisfies $0 \le X_n \nearrow X$ with $EX=\infty$, then I still could use MCT to get ...
0
votes
1answer
36 views

How are graphs drawable?

Drawing some graphs in Cartesian Coordinate system for a College Algebra class, and wondering: How are graphs even drawable? Considering that the points that make up graphs have infinitesimal size ...
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0answers
14 views

Definition of Measure regular

Book's, Real and complex analysis, Walter Rudin. I am somewhat confused. My question is: "In other words, we are looking at $L^p$, where $\mu$ is Lebesgue measure on $[0,2\pi]$(or on $T$), ...
2
votes
1answer
50 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
2
votes
2answers
43 views

$f_n \rightarrow 0$ in $L^1$ $\implies \sqrt{f_n} \rightarrow 0$ also?

Let $(X,\Sigma,\mu)$ be a finite measure space, and let $\{f_n : n \in \mathbb{N} \}$ be a sequence of non-negative measurable functions converging in the $L^1$ sense to the zero function. Show that ...
3
votes
1answer
81 views

Why will a “clock” exist a.s.?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}^2}$ and the following process: At the next time step 1 always becomes 2. 2 always becomes 0 and 0 becomes 1 if at least one of its 4 neighbours is 1, ...
2
votes
2answers
43 views

Confused about substitution in Stiltjes integral

Suppose we have an integral $$ \int_{-a}^{a} \sin (x) \nu(dx), $$ where $\nu$ is a finite measure with $\nu(-A)=\nu(A), A \in \sigma(\mathbb{R})$ and $x>0$. Then we have $$ \int_{-a}^{a} \sin (x) ...
1
vote
2answers
23 views

Measure of the intersection of a ball and a compact subset

Do you know a large class of compact subset $K$ of $\mathbb{R}^d$ such that for each such compact $K$, there exists a $r>0$ with $\inf_{x \in K} \lambda^d(B_r(x) \cap K) > 0$, where $\lambda^d$ ...
3
votes
0answers
21 views

Polish subspaces of the space of functions into a Polish space

For a topological space $E$ denote $\mathscr{B}(E)$ its Borel $\sigma$-algebra. For a measurable space $(A, \mathscr{A})$ and $B \subseteq A$ denote $\mathscr{A}|B$ as the trace $\sigma$-algebra on ...
2
votes
0answers
45 views

Development of measure and probability theory

I am interested in a reference (article, maybe a book chapter) on the development of mathematical probability theory - that is, mostly starting from the beginning of the 20th century. It is surprising ...
2
votes
1answer
46 views

Borel-measurable functions.

One definiton of Borel measurable functions is: The collection of borel measurable functions is the smallest collection of real valued functions on $\mathbb{R}$ that contains the continuous ...
1
vote
3answers
58 views

X is some random variable and f is a continuous function. Is f[E(X)] = E[f(X)]?

I am curious about at what conditions the expectation and a mapping could exchange their operation. Say, X is some random variable, and $f:R\rightarrow R$ is a continuous function. Does $$f[E(X)] = ...
1
vote
1answer
53 views

integral of lebesgue function is continuous

Let F be a lebesgue integrable function on $(0,\infty)$. For $0 \le t < \infty$, define $g(t)=\int_{0}^{\infty} e^{-tx}F(x)dx$. Can someone explain why $g$ and $g'$ are continuous over ...
0
votes
0answers
29 views

A series of measures

Let $(\mu_n)$ be a sequence of finite measures on $(\Omega,\Sigma)$. We put: $$\mu(A):= \sum\limits_{n=1}^\infty \frac{\mu_n(A)}{2^n(1+\mu_n(\Omega))}$$ Show that $\mu$ is a finite measure. I have ...
0
votes
0answers
14 views

Definition of an “arc” and possible error in proof of length of projection of regular $1$-set in $\mathbb{R}^2$.

Here is an extract from Falconer's The Geometry of Fractal Sets. I cannot see how an "arc" is defined and was wondering whether someone could help me with the definition. Also if $ \begin{align} ...
0
votes
1answer
15 views

What is meant with translation-invariant product- measure here?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}^2}$. What is meant by a tranlation-invariant product-measure on $X$? On which $\sigma$-algebra? What does translation-invariant mean here?
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0answers
36 views

How is this passage probably meant?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}^d}$. Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have ...
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1answer
32 views

Convergence in measure of sequence of functions

Hi I don't have a lot of experience in measure theory so that might be basic. If you have a sequence of functions $a_{k}(x): \Omega \rightarrow \mathbb{R}$ such that $$0 \leq\limsup\limits_{k ...
3
votes
1answer
49 views

Evaluation of Lebesgue Integral using Convergence Theorems

Using convergence theorems, I am trying to compute the value of $$ \lim_{n\to\infty}\int_a^\infty \frac n{1+n^2x^2}\,\mathbb{d}x $$ for $a \in \mathbb{R}$, and with respect to the Lebesgue measure. ...
0
votes
1answer
23 views

Bochner Integral: Approximability

Disclaimer This thread is related to: Bochner Integral: Integrability It is meant to record. See: Answer own Question It is written as jeopardy. Have fun! :) Problem Given a measure space ...
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vote
1answer
25 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
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0answers
20 views

What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability?

What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability? I know Borel measurability implies both Lebesgue-Stieltjes measurability and Lebesgue measurability. ...
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vote
2answers
80 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
5
votes
1answer
72 views

Question regarding Radon-Nikodym derivative…

The problems are as follows: (1) Let $X=[0,1]$ with Lebesuge measure and consider probability measures $\nu,\mu$ given by densities $f,g$ as follows: $$\nu(E)=\int_{E} ...
2
votes
2answers
40 views

Sets in product $\sigma$-algebras that cannot be written as a product of measurable sets in the factors.

I am aware that not every set in a product $\sigma$-algebra can be represented as a product of measurable sets in the factors (e.g., take the unit ball in $\mathbb{R}^{n}$), but this seems weird to ...
4
votes
0answers
27 views

Infinite products of non-measurable sets

I just proved for a homework problem that the direct product of two non-measurable sets is non-measurable. It seems to me that the finite direct product of finitely many non-measurable sets is also ...
4
votes
0answers
46 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...