4
votes
0answers
37 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
30 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 ...
0
votes
0answers
24 views

Measure Theory vs. Decision Theory - problem classification

I am having trouble classifying my problem, and I am seeking some guidance on book advice. I don't know if I have measure-theory problem and/or a decision-theory problem (or other field). I want to ...
1
vote
1answer
20 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
31 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
95 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 ...
0
votes
2answers
51 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
52 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
31 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
32 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
35 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 ...
2
votes
3answers
71 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
29 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
48 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
65 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
24 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 $\epsilon$ ...
0
votes
1answer
86 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
33 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
84 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
76 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
0answers
49 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 ...
4
votes
1answer
109 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, ...
2
votes
3answers
183 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
89 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 ...
4
votes
2answers
512 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
85 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
46 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
55 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
69 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
38 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 ...
6
votes
2answers
187 views

Probability measure on subset of natural numbers…

How one would define a probability measure on all subsets of natural numbers, which is finite-additive and such that the variables $\chi_p(n)=\left\{\begin{matrix} 1 & p|n \\ 0 & \text{esle} ...
3
votes
1answer
91 views

Measure dualization

What ways are known to correspond, or transfer, a Borel probability measure $\mu$ over some Banach space $X$ to a Borel probability measure $P$ over $X^{*}$, the dual space? Of course, if $X^{*} = ...
2
votes
3answers
84 views

Online reference about Geometric Measure Theory.

I would like to find an online reference about the basics of Geometric Measure Theory. The reference should treat such things as regions and isoperimetric surfaces. Can you tell me, where I can find ...
3
votes
1answer
57 views

Lebesgue density for other probability measures on $[0,1]$

Does the Lebesgue density theorem hold for arbitrary (Borel) probability measures on $[0,1]$? Following Downey & Hirschfeldt's proof leads me to believe that the answer is "yes". (Recall every ...
5
votes
1answer
130 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
4
votes
2answers
103 views

Possible typo in Bogachev's Measure Theory

I've a problem with the definition of a measure space in Bogachev's Measure Theory. The author (Definition 1.3.2, p. 9) assumes a measure to be a countably additive set function defined on an algebra ...
2
votes
0answers
72 views

General questions on measure spaces

Let $X$ be a set and $\mathcal{A}$ a sigma-algebra on $X$. Two measures are said to belong to the same class iff. they have the same null sets. Together with every measurable map between measure ...
3
votes
0answers
598 views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
2
votes
1answer
67 views

Tails of family of integrable functions

It is well known that tail of an integrable function on $\mathbb{R}^d$ is small, i.e., Given $\epsilon>0$, there is $R>0$ such that $$\int_{\{|x|>R\}}|f(x)|dx<\epsilon.$$ I was wondering ...
4
votes
2answers
92 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
2
votes
4answers
567 views

Book Suggestions for an Introduction to Measure Theory [duplicate]

Couldn't find this question asked anywhere on the site, so here it is! Do you guys have any recommendations for someone being introduced to measure theory and lebesgue integrals? A mentor has ...
0
votes
1answer
43 views

Outer measure defined by a continuous and bijective function

This problem is from K.T. Smith's Primer of Modern Analysis: Let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be continuous and one-to-one on an open set $\Omega \subset \mathbb{R}^d$ and define $$\nu(A) ...
0
votes
1answer
64 views

Sigma algebra of a regular borel measure

From the definition I am using and restrict to $\mathbb{R}^d$ only, what can we say about $\sigma$-algebra of $\nu$-measurable sets, $\mathfrak{B}_{\nu}$? Some more specefic questions: It contains ...
2
votes
1answer
120 views

Derivative of Regular Borel Measure

I just want to check if there are any other references on the definition of derivative of a regular borel measure beside the one I am reading: A regular Borel measure on an open set $\Omega \in ...
6
votes
1answer
169 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
7
votes
3answers
445 views

Is $C([0,1])$ a “subset” of $L^\infty([0,1])$?

This is motivated from an exercise in real analysis: Prove that $C([0,1])$ is not dense in $L^\infty([0,1])$. My first question is how $C([0,1])$ is identified as a subset of $L^\infty([0,1])$? ...
9
votes
2answers
109 views

Determining measures by integrals

What classes of functions are sufficient to determine whether two measures are equal? If $$\int_{R^d} f d\mu =\int_{R^d} f d\nu $$ for some functions $f$, when can we say that $\mu=\nu$? Obviously, ...
5
votes
3answers
445 views

A Good Book for Mathematical Probability Theory [duplicate]

I am from mathematical background, and I always hated the way they teach elementary probability theory in schools without giving any clue about measure theory. I want a theoretical book in ...
2
votes
1answer
335 views

Book on Measure Theoretic Statistics

I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a ...
2
votes
1answer
63 views

Uniform integrability, book

I search about this theme, in the books is as exercise. But I want some more theory. What book recommend?