Is there a non-piecewise 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 zero-dimensional 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, non-piecewise (i.e. can be described without a piece-wise 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 non-measurable 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.