Bharath Krishnan<p>I finally know what I want. </p><p>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.</p><p>§1. Motivation</p><p>Suppose, we define everywhere surjective \(f\):</p><p>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 \(V\in\mathrm{T}\).</p><p>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 \(f\):</p><p>\[\mathbb{E}[f]=\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}\]</p><p>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].</p><p>[1]: <a href="https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr" target="_blank" rel="nofollow noopener noreferrer" translate="no"><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</span></a></p><p>[2]: <a href="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" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">researchgate.net/publication/3</span><span class="invisible">89499633_Defining_a_Unique_Satisfying_Expected_Value_From_Chosen_Sequences_of_Bounded_Functions_Converging_to_an_Everywhere_Surjective_Function/stats</span></a></p><p><a href="https://mathstodon.xyz/tags/HausdorffMeasure" class="mention hashtag" rel="tag">#<span>HausdorffMeasure</span></a> <a href="https://mathstodon.xyz/tags/HausdorffDimension" class="mention hashtag" rel="tag">#<span>HausdorffDimension</span></a> <br /><a href="https://mathstodon.xyz/tags/EverywhereSurjectiveFunction" class="mention hashtag" rel="tag">#<span>EverywhereSurjectiveFunction</span></a><br /><a href="https://mathstodon.xyz/tags/ExpectedValue" class="mention hashtag" rel="tag">#<span>ExpectedValue</span></a><br /><a href="https://mathstodon.xyz/tags/Average" class="mention hashtag" rel="tag">#<span>Average</span></a> <br /><a href="https://mathstodon.xyz/tags/research" class="mention hashtag" rel="tag">#<span>research</span></a></p>