Averaging an Explicit, Non-Lebesgue Integrable, and Unbounded Function That Is Defined Without The Axiom of Choice

Motivation: The function below has an undefined mean, no matter how much we extend the Lebesgue or Hausdorff measure.

Properties of the Function: We want an explicit, unbounded $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without the axiom of choice) such that:

1. The restriction of $\mathcal{G}$ to any interval has infinite $2$-d area both above and below the $x$-axis.
2. For all real numbers $a \lt b$ and $c\lt d$, $\{x\in(a,b):\mathcal{G}(x) \in (c,d)\}$ is a set with positive Lebesgue measure.

Note, such a function does exist:

Example: Indeed, let $(q_t)_{t\in\mathbb{N}}$ be a numbering of the rational numbers and $k_t=2^{2^t}$, and define functions $s_t$ as follows:

If $s_0=0$ everywhere and:

$$ \small{ s_{t+1}(x)=\begin{cases} q_{t/2} & x\in(q_j-1/k_t,q_j+1/k_t), j=1,\cdots,t,\, t \text{ is even}\\ k_t^2 & x \in(q_j-1/k_t,q_j+1/k_t), j=1,\cdots,t,\, t \text{ is odd}\\ s_t(x) & \text{otherwise} \end{cases}}$$

each $s_{t+1}$ agrees with $s_t$ at all real numbers except a set of measure $<1/2^t$ for big $t$, so one could consider $\mathcal{G}$ the pointwise limit of the functions $s_t$, which is defined everywhere except measure $0$ (in those bad points just define $\mathcal{G}=0$). The resulting function satisfies both conditions 1. and 2. above.

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Question: How do we extend the mean of this explicit example to increase the chances of getting a finite and "satisfying" value (i.e., next section), rather than an infinite or undefined one, so that we have a more meaningful application to a real-word problem?

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I was expecting one can find their own definition of "satisfying". If not, here is my research paper.

For the sake of simplicity, I will state that the families of bounded functions are the sequences of bounded functions. Thus, the index set $\mathcal{A}(A)$ of an arbitrary family of bounded functions or sets are the natural numbers. In the paper, we set $n=1$ and $A=\mathbb{R}$. Note, the explicit example in this post is stated in pg. 3, Section 2.1.1 of this paper.

Summary of Attempt: I want to extend the mean of $\mathcal{G}$, by using a modeling question (its answer, a mathematical model of a real-life phenomenon) that takes unique and "satisfying" value, in the following order: finite, infinite, or undefined. The modeling question should include the following:

1. The "measure" (pg. 10-12, Definition 20-22) of a sequence of bounded functions converging to $\mathcal{G}$ (pg. 6, Definition 12): i.e., the "measure" partitions each bounded function's graph into equal measure sets which take a sample point from each partition, pathways of line segments between sample points, lengths of line segments in each pathway, removed lengths which are outliers, remaining lengths which are converted into a probability distribution, and the entropy of the distribution
2. A reference point $\mathbf{R}\in\mathbb{R}^{2}$
3. The actual rate (pg. 12, Definition 23) versus the expected rate of expansion of each bounded function's graph
   • The expected rate of expansion is a fixed constant.

Sketch of A Choice Function That Answers The Modeling Question: Combining “the measure” and the actual rate of expansion, we get a general notion of the choice function that answers the modeling question (pg. 13, Section 3.1). The choice function should pick a sequence of bounded functions converging to f that satisfy all criteria in the modeling question and when all other criteria in the modeling question are satisfied:

1. the greater the rate of increase of the "measure" of each bounded function's graph compared to that of other sequences of each bounded function's graph (see 1. of previous list),
2. the smaller the absolute difference between the $y$-coordinate of $\mathbf{R}$ and the mean of a chosen sequence of bounded functions converging to $f$ (pg. 7, Definition 13 and 2. of previous list),
3. the smaller the absolute difference between the expected rate of expansion and actual rate of expansion of a sequence of each bounded function's graph (see 3. of previous list),

the more likely the choice function that answers the modeling question, chooses the desired sequence. Hence, we take the mean of the chosen sequence as the new mean. (Note, the choice function actually takes multiple sequences of bounded functions converging to f such that the mean of all chosen sequences have the same "satisfying" and finite value (pg. 8, Section 2.2.4 Part 1).)

Attempt to Answer The Modeling Question: I answered the modeling question in pg. 14, Section 4. Unfortunately, the answer is too dense. I need to simplify it with code.

Question (Reasked): How do we extend the mean of the explicit example to increase the chances of getting a finite and "satisfying" value (i.e., next section), rather than an infinite or undefined one, so that we have a more meaningful application to a real-word problem?