Is my "average" of this function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ non-zero and finite?

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(I tried to make the post as self-contained as possible. If you need to read the concept paper, I refer to the exact pages needed to find the rigorous version(s) of each definition in the post.)

Organization: The post is organized in the following order,

the motivation of $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ and why the usual means of $\mathcal{G}$ do not give what I want,
the questions regarding the new mean of $\mathcal{G}$,
the summary of the concept paper which describes the new methods to averaging $\mathcal{G}$,
helpful resources for answering the questions.

Motivation: Consider the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$,

(Example of $\mathcal{G}$) Let $(q_t)_{t\in\mathbb{N}}$ be an extended Calkin-Wilf sequence of the rational numbers and $k_t=2^{2^t}$, and define functions $s_t$ as follows:

If $s_0=0$ everywhere and:
$$ \small{ s_{t+1}(x)=\begin{cases} q_{t/2} & x\in\bigcup_{j\in\{1,\cdots,t\}}(q_j-1/k_t,q_j+1/k_t),\, t \text{ is even}\\ k_t^2 & x \in\bigcup_{j\in\{1,\cdots,t\}}(q_j-1/k_t,q_j+1/k_t),\, t \text{ is odd}\\ s_t(x) & \text{otherwise} \end{cases}}$$
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 $\mathcal{G}=0$).

Here is the partial code of the example using Mathematica:

Clear["Global`*"]

enumerateRationals[n_Integer?Positive] := 
  Module[{posRationals, 
    fullList},(*Generate enough positive rationals using the Calkin-
   Wilf step*)

   posRationals = NestList[1/(2 Floor[#] - # + 1) &, 1, Ceiling[n/2]];
   (*Interleave:0,q1,-q1,q2,-q2...*)

   fullList = Riffle[posRationals, -posRationals];

   Prepend[fullList, 0][[1 ;; n]]];

enumerateRationals[0] = {};

q[t_] := q[t] = enumerateRationals[t][[t]]
(*Takes the t-value in an enumeration*)

k[t_] := k[t] = 2^(2^t)

s[0, _] = 0;
s[t_Integer, x_] := 
 s[t, x] = 
  Piecewise[{{q[(t - 1)/2], 
     EvenQ[t - 1] && 
      0 <= Min[Abs[x - enumerateRationals[t - 1]]] < 1/k[t - 1]}, {k[
       t - 1]^2, 
     OddQ[t - 1] && 
      0 <= Min[Abs[x - enumerateRationals[t - 1]]] < 1/k[t - 1]}}, 
   s[t - 1, x]]

FullSimplify[PiecewiseExpand[s[5, x]]]
(*Example of s[t,x]*)

(I want to compute s[t,x] as t->Infinity to get $\mathcal{G}(x)$ or bigG[x].)

In the usual sense, the mean of $\mathcal{G}$ w.r.t. the Lebesgue measure over a family of bounded and finite measure sets converging to $\mathrm{dom}(\mathcal{G})=\mathbb{R}$ is always undefined, since $\mathcal{G}$ satisfies two properties:

The restriction of $\mathcal{G}$ to any interval has infinite area both above and below $y=0$.
For all $a\lt b$ and $c\lt d$ real numbers, $\{x\in(a,b):\mathcal{G}(x)\in (c,d)\}$ is a set with positive Lebesgue measure.

Furthermore, the mean w.r.t. the Hausdorff measure in its dimension of a family of bounded functions (with different bounded domains of finite measure) converging to $\mathcal{G}$ is defined but has different values depending on the family of bounded functions chosen. Hence, the former mean is non-unique.

Question: In my concept paper (see the following summary to keep this post as self-contained as possible), assuming $\mathbf{R}$ is the origin and $E=1/2$, what is the average of the example of $\mathcal{G}$ using programming and analysis? Is the "average value" non-zero and finite? If not, explain why?

Every reference in the next section refers to the Sections, Definitions, and pages of the concept paper.

Summary of The Concept Paper

My attempt to average of $\mathcal{G}(x)$ on $\small{(-\infty,+\infty)}$ is defined w.r.t. to four definitions:

The reference point $\mathbf{R}\in\mathbb{R}^{2}$:

For every chosen reference point, there should be a unique, "satisfying" (i.e., see the section "Modeling Question Summary" at the bottom), and either:

a finite mean of $\mathcal{G}$

if not a finite mean of $\mathcal{G}$, then an infinite mean

if not a finite or infinite mean, then an undefined mean of $\mathcal{G}$.

The "measure" of a family of each bounded function’s graph, where $\{G_r^{\star}:r\in\mathbb{R}^{+}\}=\{\mathrm{graph}(f_r^{\star}):r\in\mathbb{R}^{+}\}$ and $\{f_r^{\star}:r\in\mathbb{R}^{+}\}$ is a family of bounded functions with different bounded domains of finite measure converging to $\mathcal{G}$):

The preliminary step to defining the "measure" of a family of each bounded function's graph (Definition 20, pg. 20-22) is to partition each of their graphs 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, the entropy of the distribution, and the maximum entropy w.r.t all pathways.

The "measure" (Definition 22, pg. 23-25) involves the supremum of the sample size (i.e., when applying the preliminary step to each bounded function's graph in a chosen family) such that the entropy in the preliminary step is less than or equal to the entropy (i.e., when applying the preliminary step to each bounded function's graph in a "non-equivalent" family [Definition 18-19, pg. 19-20]), then dividing the former by the following sample size (i.e., when applying the preliminary step to each bounded function's graph in the "non-equivelant" family) and taking its "supremum" and "infimum" w.r.t. to all possible partitions and samples. (The "supremum" and "infimum" should be equal.) To better understand the definitions, consider the following examples (Section A.7.1-A.7.3, pg. 179-197).

The expected rate of expansion:

The expected rate of expansion is an arbitrary fixed constant and is written as the function $E:\mathcal{A}(A)\to\mathbb{R}$ (i.e., $\mathcal{A}(A):=\mathbb{R}^{+}$ is the chosen index set of a family of functions or sets).

The actual rate of expansion:

The actual rate of expansion of a family of each bounded function's graph is the "derivative" of a function of the "average" $2$-dimensional Euclidean distance between every point in each bounded function's graph and the reference point $\mathbf{R}\in\mathbb{R}^{2}$ (Definition 23, pg. 25-26)

Modeling Question Summary: Combining "the measure" and the actual rate of expansion, we get a general notion of the choice function in the modeling question (Section 3.1, pg. 27-30). The choice function should pick "equivelant" families (Definition 15-17, pg. 17-18) of bounded functions converging to $\mathcal{G}$ which satisfy all the criteria in the modeling question, such that:

the "measure" of each chosen family of each bounded function's graph increases at a rate linear or superlinear (Definition 22, pg. 23-25) to that of each "non-equivelant" family (Definition 18-19, pg. 19-20) of each bounded function's graph (Section 3.1 criterion 2)
the absolute difference between the $y$-coordinate of the reference point $\mathbf{R}\in\mathbb{R}^{2}$ and the mean of each chosen family of bounded functions converging to $\mathcal{G}$ is minimized (Definition 13 [pg. 13-14], Section 3.1 criterion 4)
the absolute difference between the expected rate of expansion and the actual rate of expansion of each chosen family of each bounded function's graph is minimized (Definition 23 [pg. 25-26], Section 3.1 criterion 4), so that it's more likely the choice function, which answers the modeling question, chooses the desired families. Hence, we take the mean of the chosen "equivelant" (Definition 15-17, pg. 17-18) families as the new mean (i.e., the "equivelant" families should have the same mean).

Useful Resources:

Since I was not formally trained in Mathematica, I asked smaller questions to get answers to the bigger ones. Here is the progress I made over the years.

Defining a "pathway" from a list of points

The above post takes the preliminary step to applying the "measure". (However, the preliminary step in the former post was applied to sets rather than the graphs of functions.)

Plotting/evaluating a function defined from a piecewise recurrence relation

The above post (once answered) helps visualize and understand the function $\mathcal{G}$. This could help answer the question, assuming there is symmetry in the function $\mathcal{G}$.

Answer from ReserachGate and its Mathematica code

In the above links, a researcher gave two answers to the post on ResearchGate. Since the researcher might have used AI, I'm hesitant to accept these answers. (I hope someone can check whether the researcher is correct.)

If the answers are wrong, some of the steps can still be useful.

Communities

Is my "average" of this function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ non-zero and finite?

Summary of The Concept Paper

Useful Resources:

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