3
votes
1answer
40 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
2
votes
1answer
26 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = ...
1
vote
1answer
43 views

Reference request: Measure theory and/or manifolds [duplicate]

I have never encountered measure theory or manifolds yet, despite being close to my third year university level. Any texts for either or both of these subjects would be greatly appreciated.
1
vote
1answer
39 views

Extension of measures

Here an exercise of the book: A probability path by Sidney Resnick. Suppose $P$ is a probability in a $\sigma$-field $\cal{B}$ and $A\notin \cal{B}$. Let $\mathcal{B}_1:=\sigma(\mathcal{B}\cup\{A\})$ ...
1
vote
0answers
56 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
2
votes
2answers
110 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
6
votes
2answers
59 views

Existence of a random variable satisfying a condition on its distribution

Let $X, Y : [0,1] \to \mathcal{X}$ be two random variables. Here, $[0,1]$ is the interval with the Lebesgue $\sigma$-algebra and $\mathcal{X}$ is a topological space with the Borel $\sigma$-algebra. ...
1
vote
1answer
106 views

Measure Theory Book

What book should I use for measure theory?I have solved Rudin's Principle Of mathematical analysis up to chapter 7.Some people advised me to use Real and complex analysis by Rudin, while other said it ...
0
votes
0answers
13 views

Is every frame homomorphism induced by a measurable function?

Let $M$ be the Lebesgue measure algebra of the unit interval $[0,1]$, i.e. equivalence classes of Lebesgue measurable sets modulo sets of measure $0$. This is a complete Boolean algebra, hence in ...
0
votes
1answer
41 views

Counterexamples in measure theory

Can you suggest me a book which primarily deals with counter-examples in measure theory? Thank You in advance!
0
votes
1answer
34 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
0
votes
1answer
33 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
3
votes
2answers
102 views

Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure ...
2
votes
3answers
86 views

Recommendations for books on complex analysis and on measure theory?

I'm looking for a book on complex analysis that has a similar writing style to either Terry Tao's Analysis II or Nathan Jacobson's Basic Algebra series. I have found both of these extremely easy to ...
0
votes
0answers
28 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
5
votes
1answer
144 views

Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
1
vote
0answers
23 views

Limit a.s. of a sequence of normal random variables is normal.

I know that the statement "If $X_n$ is a sequence of normal random variables which converges a.s. to a random variable $X$, then $X$ is also a normal random variable" is true. However, do you ...
14
votes
2answers
375 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
1
vote
1answer
39 views

Kolmogorov's Existence Theorem

My analysis professor told us to take the following theorem for granted in order to prove other results, but I would like to see a proof of it, since I think it will be beneficial. Here is the ...
0
votes
1answer
63 views

Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
4
votes
0answers
94 views

Do most nowhere dense sets have measure $0$?

Inspired by this question here and in particular the answer I was wondering: Do most nowhere dense sets have measure zero? By "most" I mean in the sense of the "measure" of the set of all nowhere ...
5
votes
0answers
57 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
1
vote
1answer
50 views

References for a second course in probability theory

I need a probability book that treats all the arguments from the point of view of the measure theory and the Lebesgue integral. I've the basis of "naive" probability theory and of measure theory so I ...
2
votes
1answer
37 views

Lebesgue-Radon-Nikodym Theorem without Hilbert spaces

In my analysis class we are seeing the so called Lebesgue-Radon-Nikodym Theorem. But we prove it the "old fashioned way" without using Hilbert space theory. More precisely, we prove the minimality ...
2
votes
2answers
32 views

Various “sizes” of 0-measured sets

I am looking for a formalization of an intuitive concept of size, in cases simple measure is too coarse. It will be easier for me to give an example. Let $\mu$ be the Lebesgue measure on the unit ...
0
votes
3answers
344 views

Measure Theory and Functional analysis exercise book

I'm looking for a big collection of exercises of functional analysis and measure theory. I know a lot of theory books which present some excercises (Brezis, Rudin, Lang, Royden, and others) but I was ...
1
vote
2answers
110 views

Books on measure theory

Can someone kindly suggested a good book on measure theory? Taking into consideration a good treatment of the abstract measures and Caratheodory approach.
1
vote
1answer
85 views

Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
0
votes
1answer
37 views

Is there any good text introducing a part of Borel-hierarchy which is in need in measure theory

Is there any good text introducing a subpart of Borel-hierarchy which is in need in measure theory, which can be done in short time? Say, 1~3 days if possible. (Assuming i'm studying about 14hours a ...
1
vote
1answer
53 views

Reference for a proof of which 2-increasing functions are joint cdf's

Can somebody give me a reference giving the detailed statement and proof of the fact that the joint cdf's of positive Borel measures $\mu$ on $\mathbb{R}^2$, so $$F(a,b) = \mu(\{(x,y) : x \leq a, y ...
0
votes
1answer
77 views

Hausdorff content and Hausdorff measure

I am dealing with the Hausdorff dimension and I came across two different ways of defining this dimension. This question is possibly related to Hausdorff Measure and Hausdorff Dimension but the ...
1
vote
3answers
76 views

Recommend me a text or webpage introducting gamma function throughly

Till now, i have learned abstract Integration, all basic properties of the (n-dimensional) Lebesgue(-Stieltjes) measure and the lebesgue integral is an extension of Riemann integral. Here's an ...
1
vote
0answers
40 views

Preimage of zero measure sets

Let $A\subset\mathbb{R}$ be a set of Lebesgue measure zero and $f:\mathbb{R}^n\to\mathbb{R}$ is a function. Under what conditions does $f^{-1}(A)$ have Lebesgue measure zero? I found a possible ...
1
vote
0answers
51 views

need help with choosing a book

Does anyone know a good general measure book that has good examples and theories. I tried reading Royden real analysis but I need more books to help me understand the material. any help is greatly ...
3
votes
1answer
131 views

Definition of Liouville measure on energy surface of Hamiltonian system

This is a reference request, as I can't for the life of me find anything that answers my question in the literature. If $(M,\omega,H)$ is a Hamiltonian system, we know from Liouvile's theorem that ...
1
vote
0answers
27 views

Gaussian Smoothing Error and “Hard Analysis” Bounds

Let $p \in (0,\infty)$. Consider a function $f \in L^p([0,1])$, and let $$\phi_\epsilon(x) = \frac{\exp(-x^2/2\epsilon^2)}{\sqrt{2\pi\epsilon^2}}$$ denote a $0$-mean Gaussian of variance $\epsilon$ ...
1
vote
1answer
120 views

Banach Measures: total, finitely-additive, isometry invariant extensions of Lebesgue Measure

I've been reading about paradoxical sets, mainly paradoxical subsets of the plane. As a consequence of this, I've been reading a couple of G.A. Sherman's papers on the subject. In his paper ...
4
votes
0answers
45 views

How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
0
votes
1answer
246 views

Equality in Minkowski's theorem

I would like to see a proof of when equality holds in Minkowski's inequality. The proof is quite different for when $p=1$ and when $1<p<\infty$. Could someone provide a reference? Thanks!
5
votes
0answers
80 views

Is $\mathbb{R}/\mathbb{Q}$ an interesting group? [duplicate]

Inspired by the construction of the non-mesurable Vitali set I thought about the group $\mathbb{R}/\mathbb{Q}$ (the additive group of the real numbers modulo the rationals). There must be some ...
3
votes
1answer
68 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
5
votes
1answer
164 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
3
votes
3answers
228 views

Any suggestions about good Analysis Textbooks that cover the following topics?

I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's ...
2
votes
1answer
134 views

Lyapunov Theorem for beginners

I study the subject of fair division (cake-cutting), and many papers contain a reference to a theorem by Lyapunov, which states that the range of any real-valued, non-atomic vector measure is compact ...
5
votes
5answers
774 views

Measure theory and topology books that have solution manuals

I am trying to find a book to learn measure theory that contains complete solutions manual. Does someone know of any? Also, I would like to know if there is a book with solutions manuals about ...
2
votes
0answers
96 views

Textbook Recommendation; Proability Theory with Measure Theory

I'm currently taking a course in Probability Theory and was hoping someone could point me in the direction of a useful supplementary textbook. Our course currently uses A Modern Approach to ...
2
votes
0answers
47 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
2
votes
1answer
63 views

Book searching in Pluripotential theory

Can anyone recommend me a book on pluripotential theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why ...
3
votes
1answer
75 views

Extend a linear functional on “nice” functions to an integral

I have a positive linear functional $h$ defined on a set of Lesbesgue-measurable functions of "moderate growth" on $\mathbb{R}^2$–call this set $MG(\mathbb{R}^2)$. (A function $f$ is positive if ...
0
votes
0answers
44 views

Formulating rigorously the notion of “countable infinite many set-theoretic operations”

For algebras generated by a family of sets there is a way to explicitly describe how this algebra looks like. But my measure-theory lecture notes tell me, that for $\sigma$-algebras this is not the ...