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].
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