Preliminaries

Consider the expectations described in this paper, which is an extension of the Lebesgue density theorem; this paper, which is an extension of the Hausdorff measure using Hyperbolic Cantor sets; and this paper, which applies a Henstock-Kurzweil type integral on a measure Metric Space. We could also use conditional expectation; however, the output depends on the choice of the "condition". Moreover, we assume there is no "known" choice function which chooses conditions which "naturally" extend expected values for the previous sentences to be unique and finite.

Motivation

According to an article in Quanta Magazine Wood writes, "No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe."

I want to represent this with functions—I'm looking for two examples:

1. A function which matches Wood's description, such the expected values w.r.t any measure in the preliminaries is infinite or undefined.

2. A "non-fractal" function where none of the expected values in the preliminaries gives a unique, finite, expected value.

One example of 2. is $f:\mathbb{Q}\to\mathbb{R}$, where:

\begin{equation} f(x)=\begin{cases} 1 & x\in A_1:=\left\{r/q:r\in\text{odd }\mathbb{Z},q\in\text{even }\mathbb{Z},q\neq 0,\gcd(r,q)=1\right\}\\ 0 & x\in A_2:=\left\{r_1/(q_1):r_1\in\mathbb{Z},q_1\in\text{odd }\mathbb{Z},\gcd(r_1,q_1)=1\right\} \end{cases} \end{equation}

we could find a unique average using conditional expectation of $f$ given a sequence of sets with a set-theoretic limit of $\mathbb{Q}$; however, the conditional expectation depends on the sequence chosen. (Hence, the expected value could be any value and is undefined.)

Questions: Does there exist an explicit function which answers 1. and 2.?

This post might be able to help.