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Motivation: According to this question

Some sets have a Hausdorff Dimension $\alpha$ but have a zero-dimensional Hausdorff Measure. These sets may have another 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 gauge function since the sets are either $\sigma$-finite with respect to the counting measure (i.e. Countably infinite sets) or their gauge function does not exist (see the examples in Dave L. Renfro's answer).

Despite this, I need a unique and natural extension of the Hausdorff Measure to be positive and finite for sets with no meaningful gauge function (see the previous paragraph) and for all “nice” sets (i.e. sets in the $\sigma$-algebra of Caratheodory measurable sets) so the average of the function (i.e. the integral of function w.r.t to the extended measure of the domain, divided by the extended measure of the domain) exists.

Question:

How do we naturally extend the average from the Hausdorff Measure and Integral w.r.t the Measure to exist for all functions with a domain without a gauge function but in the $\sigma$-algebra of Caratheodory-Measurable sets?

More specifically, the extended average must be unique and defined between the infimum and supremum of the function we are averaging over.

Does such an average exist?

Edit: Potential Answers

See Integration with Filters. This appears to go after the problem stated but the average isn’t unique as is with most integration which use filters. Despite this, there exists a unique minimal filter (in Theorem 7) where its average is compatible with the Hausdorff Measure.

Edit 2: Another answer is from Non-Standard Measure Theory: Hausdorff Measure. This again uses filters and extends the Hausdorff Measure; however, the extension isn’t unique.

Edit 3: Perhaps we could create a choice function which chooses a specific minimal filter for the filter integral (see Theorem 7 of Integration with Filters) so the average from the filter integral, for functions with domains that have a gauge function, matches the average from the Hausdorff Measure and Integral as well as give a unique average for functions with a domain with no gauge function but in the $\sigma$-algebra of Caratheodory-measurable sets.

Question 2: Does such a choice function which is natural exist? What is an example?

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    $\begingroup$ For what it's worth, my correct legal (birth) first name is "Dave", not "David", although I often wind up seeing "David" written (sometimes even on official documents) because quite often people think in my case that "Dave" is a nickname (which is usually the case when someone goes by "Dave", so it's quite understandable). $\endgroup$ Feb 28 at 19:49

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