I finally know what I want.

Let \(n\in\mathbb{N}\) and suppose function \(f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}\), where \(A\) and \(f\) are Borel. Let \(\text{dim}_{\text{H}}(\cdot)\) be the Hausdorff dimension, where \(\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)\) is the Hausdorff measure in its dimension on the Borel \(\sigma\)-algebra.

§1. Motivation

Suppose, we define everywhere surjective \(f\):

Let \((A,\mathrm{T})\) be a standard topology. A function \(f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}\) is everywhere surjective from \(A\) to \(\mathbb{R}\), if \(f[V]=\mathbb{R}\) for every \(V\in\mathrm{T}\).

If f is everywhere surjective, whose graph has zero Hausdorff measure in its dimension (e.g., [1]), we want a unique, satisfying [2] average of \(f\), taking finite values only. However, the expected value of \(f\):

\[\mathbb{E}[f]=\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}\]

is undefined since the integral of \(f\) is undefined: i.e., the graph of \(f\) has Hausdorff dimension \(n+1\) with zero \((n+1)\)-dimensional Hausdorff measure. Thus, w.r.t a reference point \(C\in\mathbb{R}^{n+1}\), choose any sequence of bounded functions converging to \(f\) [2, §2.1] with the same satisfying [2, §4] and finite expected value [2, §2.2].

[1]: https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr

[2]: https://www.researchgate.net/publication/389499633_Defining_a_Unique_Satisfying_Expected_Value_From_Chosen_Sequences_of_Bounded_Functions_Converging_to_an_Everywhere_Surjective_Function/stats

#HausdorffMeasure #HausdorffDimension
#EverywhereSurjectiveFunction
#ExpectedValue
#Average
#research

Question 1. was solved here [1]. The answer isn't perfect but it's better than nothing.

I don't know if the function in the answer to this post [1] has a finite expected value using section 3.2 and 6.1 of this paper [2].

[1]: https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr/476609#476609

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

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

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