Is there a nonpiecewise average which extends the average from the Hausdorff Measure and Integral and exists for all functions with a domain in the $\sigma$field on the Caratheodory extension?
Motivation:
Some sets have a Hausdorff Dimension $\alpha$ but have a zerodimensional Hausdorff Measure. These sets may have another dimension function https://en.m.wikipedia.org/wiki/Dimension_function i.e. a function $h:[0,\infty]\to[0,\infty]$ such that if we change the definiton of the Hausdorff Measure by replacing $R^{\alpha}$ with $h(R)$ (where $R$ denotes the radius of a ball in covering), the value of the Hausdorff Measure is positive and finite.
Unfortunately, there are sets with no meaningful dimension function since the sets are either $\sigma$finite with respect to the counting measure (i.e. Countably infinite sets) or their dimension function does not exist.
Despite this, I need an extension of the Hausdorff Measure to be positive and finite (for the cases mentioned above) so the average from the extended measure and Integral exists.
Question:
How do we extend the average from the Hausdorff Measure and Integral to exist for all functions with a domain in the $\sigma$algebra in the Caratheodory extension?
More specifically, the extended average must be unique, nonpiecewise (i.e. can be described without a piecewise function), and defined between the infimum and supremum of the function we are averaging over.
Does such an average exist?

My intuition says that, for any meaningful measure you can define, the Axiom of Choice should allow for the construction of a nonmeasurable set wrt to that measure. You can get rid of AC and replace it with an axiom that says that there exists a measure function defined on all the subsets of R^n, but it won't be constructive: you won't get a definition, just the existence.

The word "extend" is doing a lot of work here. To what extent do we want this average to agree with the one induced by the Hausdorff measure?

I want this average to include the one induced by the Hausdorff Measure and give a unique and satisfying value for cases where an exact dimension function (or gauge function) doesn’t exist.
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