Bharath Krishnan<p>Question 1. was solved here [1]. The answer isn't perfect but it's better than nothing.</p><p>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].</p><p>[1]: <a href="https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr/476609#476609" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">mathoverflow.net/questions/476</span><span class="invisible">471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr/476609#476609</span></a></p><p>[2]: <a href="https://www.researchgate.net/publication/382557954_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_the_Most_Pathalogical_Functions" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">researchgate.net/publication/3</span><span class="invisible">82557954_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_the_Most_Pathalogical_Functions</span></a></p><p><a href="https://mathstodon.xyz/tags/Answer" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Answer</span></a> <a href="https://mathstodon.xyz/tags/PathalogicalFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>PathalogicalFunctions</span></a> <a href="https://mathstodon.xyz/tags/EverywhereSurjectiveFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>EverywhereSurjectiveFunctions</span></a> <a href="https://mathstodon.xyz/tags/Mean" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Mean</span></a> <a href="https://mathstodon.xyz/tags/ExpectedValue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>ExpectedValue</span></a> <a href="https://mathstodon.xyz/tags/MeasureTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>MeasureTheory</span></a> <a href="https://mathstodon.xyz/tags/HausdorffMeasure" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>HausdorffMeasure</span></a> <a href="https://mathstodon.xyz/tags/HausdorffDimension" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>HausdorffDimension</span></a></p>
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#pathalogicalfunctions
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Bharath Krishnan<p>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.</p><p>Question: If \(G\) is the graph of \(f\), is there an explicit \(f\) such that:<br>1. The function \(f\) is everywhere surjective (i.e., \(f[(a,b)]=\mathbb{R}\) for all non-empty open interval \((a,b)\))<br>2. \(\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=0\)</p><p>If such \(f\) exists, we denote this special case of \(f\) as \(F\).</p><p>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 \(\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=+\infty\).</p><p>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? </p><p>If not, see this paper [2] for a partial solution. </p><p>Optional: Is there other interesting properties of \(F\)?</p><p>[1]: <a href="https://en.wikipedia.org/wiki/Conway_base_13_function" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Conway_b</span><span class="invisible">ase_13_function</span></a></p><p>[2]: <a href="https://www.researchgate.net/publication/382557954_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_the_Most_Pathalogical_Functions/stats" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">researchgate.net/publication/3</span><span class="invisible">82557954_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_the_Most_Pathalogical_Functions/stats</span></a></p><p><a href="https://mathstodon.xyz/tags/PathalogicalFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>PathalogicalFunctions</span></a> <a href="https://mathstodon.xyz/tags/EverywhereSurjectiveFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>EverywhereSurjectiveFunctions</span></a> <a href="https://mathstodon.xyz/tags/Mean" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Mean</span></a> <a href="https://mathstodon.xyz/tags/ExpectedValue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>ExpectedValue</span></a> <a href="https://mathstodon.xyz/tags/MeasureTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>MeasureTheory</span></a> <a href="https://mathstodon.xyz/tags/Measure" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Measure</span></a> <a href="https://mathstodon.xyz/tags/HausdorffMeasure" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>HausdorffMeasure</span></a> <a href="https://mathstodon.xyz/tags/HausdorffDimension" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>HausdorffDimension</span></a></p>
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