Measure And Derivative A Unified Approach by G.E. Shilov; B.L. Gurevich

This volume is intended as a textbook for students of
mathematics and physics, at the graduate or advanced
undergraduate level. It should also be intelligible to
readers with a good background in advanced calculus
and sufficient “mathematical maturity.”
The phrase “unified approach” in the title of the book
refers to the consistent use of the Daniell scheme, which
starts from the concept of an elementary integral defined
(axiomatically) on a family of elementary functions. In
the Introduction we explain in detail why we prefer
this approach to others, in particular to the Lebesgue-
Radon-Frechet approach, which starts from axiomatic
measure theory.

Revised English Edition
Translated and Edited by Richard A. Silverman

You can get the book here and here.

#1966 #derivative #higherMathematics #integral #lebesgueIntegral #LeviSTheorem #mathematics #measureTheory #physics #RiemannIntegral #sovietLiterature #StieltjesIntegral #theoryOfIntegral

I was told to share this motivational meme I made today.

#measuretheory #maths

`Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an "if and only if" characterisation of when two such measures are equivalent, and hence it is extremely useful when trying to establish change-of-measure formulae for measures on function spaces.`

https://en.wikipedia.org/wiki/Kakutani's_theorem_(measure_theory)

#measureTheory #math #mathematics #integrability

Suppose \(A\subseteq\mathbb{R}^{2}\) is Borel and \(B\) is a rectangle of \(\mathbb{R}^2\). In addition, suppose the Lebesgue measure on the Borel \(\sigma\)-algebra is \(\lambda(\cdot)\):

Question: How do we define an explicit \(A\), such that:
1. \(\lambda(A\cap B)>0\) for all \(B\)
2. \(\lambda(A\cap B)\neq\lambda(B)\) for all \(B\)?

For a potential answer, see this reddit post [1]. (It seems the answer is correct; however, I wonder if there's a simpler version that is less annoying to prove.)

Moreover, we meaningfully average \(A\) with the following approach:

Approach: We want an unique, satisfying extension of the expected value of \(A\), w.r.t the Hausdorff measure in its dimension, on bounded sets to \(A\), which takes finite values only

Question 2: How do we define "satisfying" in this approach?

(Optional: See section 3.2, & 6 of this paper [2].)

[1]: https://www.reddit.com/r/mathematics/comments/1eedqbx/is_there_a_set_with_positive_lebesgue_measure_in/

[2]: https://www.researchgate.net/publication/382994255_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_The_Most_Pathalogical_Sets

#UnboundedSets #Sets #LebesgueMeasure #MeasureTheory #Measure #ExpectedValue #Expectancy #Mean #Integration #HausdorffMeasure #HausdorffDimension

Suppose \(f:\mathbb{R}\to\mathbb{R}\) is Borel. Let \(\text{dim}_{\text{H}}(\cdot)\) be the Hausdorff dimension and \(\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)\) be the Hausdorff measure in its dimension on the Borel \(\sigma\)-algebra.

Question: If \(G\) is the graph of \(f\), is there an explicit \(f\) such that:
1. The function \(f\) is everywhere surjective (i.e., \(f[(a,b)]=\mathbb{R}\) for all non-empty open interval \((a,b)\))
2. \(\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=0\)

If such \(f\) exists, we denote this special case of \(f\) as \(F\).

Note, not all everywhere surjective \(f\) satisfy criteria 2. of the question. For example, consider the Conway base-13 function [1]. Since it's zero almost everywhere, \(\text{dim}_{\text{H}}(G)=1\), and \(\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=+\infty\).

Question 2: For any real \(\mathbf{A},\mathbf{B}\) is the expected value of \(\left.f\right|_{[\mathbf{A},\mathbf{B}]}\), w.r.t the Hausdorff measure in its dimension, defined and finite?

If not, see this paper [2] for a partial solution.

Optional: Is there other interesting properties of \(F\)?

[1]: https://en.wikipedia.org/wiki/Conway_base_13_function

[2]: https://www.researchgate.net/publication/382557954_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_the_Most_Pathalogical_Functions/stats

#PathalogicalFunctions #EverywhereSurjectiveFunctions #Mean #ExpectedValue #MeasureTheory #Measure #HausdorffMeasure #HausdorffDimension

Every irrational number has a unique infinite continued fraction expansion.

Consider the subset whose continued fraction coefficients has an infinite subsequence with each term dividing the next.

This is an explicit non-Borel measurable set, described by Lusin in 1927

#measuretheory

New almostsure blog post: Non-measurable sets

#measuretheory

https://almostsuremath.com/2023/07/16/non-measurable-sets/

A third (!) amazing thing I’ve learned today!

(This one’s a bit less accessible than the other two, as it requires some mathematical knowledge)

There is, it seems, a strictly monotone polynomial-valued measure for subsets of R^n

Video: https://youtube.com/watch?v=h_CFMtRQiek

Preprint: https://arxiv.org/abs/2008.09969

#mathematics #MeasureTheory #topology