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