What is an applicable way of averaging an everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure?

Suppose $f:\mathbb{R}\to\mathbb{R}$ is an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with a zero $2$-d Hausdorff measure.

Since the integral of $f$ w.r.t. $2$-d Hausdorff measure is undefined, the expected value of $f$ w.r.t. the $2$-d Hausdorff measure is undefined.

Thus, we take the mean of a sequence of bounded functions with different domains converging to $f$ (when it exists). For the sake of application, we want the mean to be finite.

The problem is depending on the sequence of bounded functions chosen, the expected value of the sequence of bounded functions can be one of several values (when it exists). Infact, the set of all $f$ where the expected value of two sequences of bounded functions converging to $f$ have non-equivelant expected values (when either exist), forms a prevelant “full measure” subset of $\mathbb{R}^{\mathbb{R}}$.

Hence, we need a useful way of choosing a “satisfying” expected value. There are many ways but one involves an answer to a leading question (i.e., using a choice function) with applications in physics.

For instance, the leading question can be defined w.r.t. four criteria:

1. the chosen sequences of bounded functions, which converge to an arbitrary $\mathsf{f}\in\mathbb{R}^{\mathbb{R}}$, have the same finite expected value. (This means that the chosen sequences are equivelant to each other. However, when there exists a $\mathsf{f}\in\mathbb{R}^{\mathbb{R}}$ where a sequence of bounded functions converging to $\mathsf{f}$ has a non-equivelant expected value, the sequence is non-equivelant to the chosen sequences. Moreover, if one chosen sequence out of all the chosen sequences satisfy a criteria, then so do all other chosen sequences.)
2. If 1. is true, the metric entropy (see Edit 1 at the end) of the chosen sequence of each bounded function's graph increases at a rate linear or superlinear to compared to that of all "non-equivelant" sequences of each bounded function's graph (i.e., the chosen "non-equivelant" sequences of bounded functions satisfy 1.).
3. If 1. and 2. is true, the absolute difference between the 2nd coordinate of the reference point $\mathbf{R}\in\mathbb{R}^{2}$ and the expected value of the chosen sequence of bounded functions converging to $f$ is minimized w.r.t. the same measurement of all chosen sequences of bounded functions satisfying 1. and 2.
4. If 1., 2., and 3. is true, the absolute difference between the expected rate of expansion and the actual rate of expansion of the chosen sequence of each bounded function's graph is minimized w.r.t. the same measure of all chosen sequences of each bounded function's graph (i.e., the chosen "non-equivelant" sequences of bounded functions satisfy 1., 2., and 3.).

Question: Similar to my attempt, how does one define a leading question that chooses a "satisfying" and finite average for an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure. (Use the four criteria.)

Edit 1: If the post is still unclear, see my original paper (i.e., "families" are a generalization of sequences and the metric entropy is assumed to be the "measure"). If the assumptions are incorrect, please correct me.

posted 18 days ago

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17d ago

bharathk98‭

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