Defining a unique, satisfying expected value from chosen sequences of bounded functions converging to an everywhere surjective function

Let $n\in\mathbb{N}$ and suppose function $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$, where $A$ and $f$ are Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, where $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.

$\S$1. Motivation

Suppose, we define everywhere surjective $f$:

Let $(A,\mathrm{T})$ be a standard topology. A function $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ is everywhere surjective from $A$ to $\mathbb{R}$, if $f[V]=\mathbb{R}$ for every $V\in\mathrm{T}$.

If $f$ is everywhere surjective, whose graph has zero Hausdorff measure in its dimension (e.g., [1]), we want a unique, satisfying ($\S$4) average of $f$, which takes finite values only. However, the expectation of $f$:

$$\mathbb{E}[f]=\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}$$

is undefined due to division by zero: i.e., $\mathcal{H}^{\text{dim}_{\text{H}}(A)}(A)=0$. Therefore, w.r.t a reference point $C\in\mathbb{R}^{n+1}$($\S$4.1), choose every sequence of bounded functions converging to $f$($\S$2.1) with the same satisfying ($\S$4) and finite expected value ($\S$3.2).

Note, with this paper [2], we gain more insight into the motivation.

$\S$2. Preliminaries

$\S$2.1. Preliminary Definition (Sequences of Functions Converging to $f$)

A sequence of functions $(f_r)_{r\in\mathbb{N}}$, where $f_{r}:A_r\to\mathbb{R}$and$(A_r)_{r\in\mathbb{N}}$ is a sequence of sets, converges to $f$ when:

For any $x\in A$, there exists a sequence $\mathbf{x}\in A_r$s.t.$\mathbf{x}\to(x_1,\cdot\cdot\cdot,x_n)$ and $f_r(\mathbf{x})\to f(x_1,\cdot\cdot\cdot,x_n)$.

This is equivelant to:

$(f_r,A_r)\to (f,A)$

$\S$2.2. Preliminary Definition (Expected Value of Sequences of Functions Converging to $f$)

Suppose, we define:

$(f_r,A_r)\to(f,A)$
$|\cdot|$ is the absolute value
the integral is defined, w.r.t the Hausdorff measure in its dimension

The expected value of $(f_r)_{r\in\mathbb{N}}$ is a real number $\mathbb{E}[f_r]$, when the following is true:$$\small{\begin{align}& \forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A_r)}\left(A_r\right)}\int_{A_r}f_{r}\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A_r)}-\mathbb{E}[f_r]\right|< \epsilon\right)\end{align}}$$otherwise, when no such $\mathbb{E}[f_r]$ exists, $\mathbb{E}[f_r]$ is infinite or undefined.

$\S$2.3. Preliminary Definition (Prevelant and Shy Sets)

Here is the source [3, def. 3.1]:

A Borel set $E\subset X$ is said to be prevalent if there exists aBorel measure $\mu$ on $X$ such that:
$0<\mu(C)<\infty$ for some compact subset $C$ of $X$, and
the set $E+x$ has full $\mu$-measure (that is, the complement of $E+x$ has measure zero) for all $x\in X$.
More generally, a subset $F$ of $X$ is prevalent if $F$ contains a prevalent Borel Set.

Additionally:

The complement of a prevelant set is a shy set.

Thus, notice:

If $F\subset X$ is prevelant, we say almost every element of $X$ lies in $F$.
If $F\subset X$ is shy, we say almost no element of $X$ lies in $F$.

$\S$3. Motivation to Answer $\S$1

$\S$3.1. Problems

If $\mathbb{E}[f]$ is the expected value of $f$, w.r.t the Hausdorff measure in its dimension,

$$\mathbb{E}[f]=\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A)}$$

then suppose:

$\mathbf{B}(\cdot)$ is the set of all bounded Borel functions in a function space
$\mathfrak{B}(\cdot)$ is the set of all bounded Borel subsets of a set

therefore:

If $f$ is everywhere surjective ($\S$1), whose graph has zero Hausdorff measure in its dimension (e.g., [1]), $\mathbb{E}[f]$ is undefined and non-finite.
If $F\subset\mathbb{R}^{A}$ is the set of all $f\in\mathbb{R}^{A}$, where $\mathbb{E}[f]$ is finite, then $F$ is shy ($\S$2.3).
For all $r,v\in\mathbb{N}$, suppose $f_r,g_v\in\mathbf{B}(\mathbb{R}^{A})$ and $A_r,B_v\in\mathfrak{B}({\mathbb{R}^{n}})$. If $F\subset\mathbb{R}^{A}$ is the set of all $f\in\mathbb{R}^{A}$, where $(f_r,A_r),(g_v,B_v)\to(f,A)$($\S$2.1) and $\mathbb{E}[f_r]\neq\mathbb{E}[g_v]$($\S$2.2), then $F$ is prevalent ($\S$2.3).

$\S$3.2. Approach to Solving the Statements In $\S$3.1

To solve the statements in $\S$3.1 at once, consider the following:

Suppose $\mathcal{B}\subset\mathbf{B}(\mathbb{R}^{A})$ and $\mathscr{B}\subset\mathfrak{B}({\mathbb{R}^{n}})$ are arbitrary sets, where for all $r\in\mathbb{N}$, $f_r\in\mathcal{B}$ and $A_r\in\mathscr{B}$. If $F\subset\mathbb{R}^{A}$ is the set of all $f\in\mathbb{R}^{A}$, where $(f_r,A_r)\to(f,A)$($\S$2.1) and $\mathbb{E}[f_r]$($\S$2.2) is unique, satisfying ($\S$4), and finite, then $F$ should be:
a prevalent ($\S$2.3) subset of $\mathbb{R}^{A}$
If not prevalent ($\S$2.3) then neither a prevalent nor shy ($\S$2.3) subset of $\mathbb{R}^{A}$?

Question: How do we define "satisfying" in $\S$3.2, w.r.t a reference point $C\in\mathbb{R}^{n+1}$, so that $\mathbb{E}[f_r]$ satisfies $\S$1? (In $\S$4, we give a partial solution to this question, where $(f_r)_{r\in\mathbb{N}}=(f_r^{\star})_{r\in\mathbb{N}}$.)

$\S$4. Example of Partial Solution Explaining The Term "Satisfying" In $\S$1 and $\S$3.2

We ask a leading question in $\S$4.2 with a answer that should solve the question in $\S$3.2.

$\S$4.1. Preliminaries.

Suppose, there exists arbitrary sets $\mathcal{B}\subset\mathbf{B}(\mathbb{R}^{A})$ and $\mathscr{B}\subset\mathfrak{B}({\mathbb{R}}^{n})$($\S$3.1), where for all $r,v\in\mathbb{N}$:

$f_r^{\star}\in\mathcal{B}$ and $A_r^{\star}\in\mathscr{B}$
$f_{v}^{\star\star}\in\mathbf{B}(\mathbb{R}^{A})\setminus\mathcal{B}$ and $A_{v}^{\star\star}\in\mathfrak{B}({\mathbb{R}}^{n})\setminus\mathscr{B}$
$(G_{r}^{\star})_{r\in\mathbb{N}}=(\text{graph}(f_{r}^{\star}))_{r\in\mathbb{N}}$ is the sequence of the graph of each $f_r^{\star}$
$\square$ is the logical symbol for "it's necessary"
$C$ is a reference point in $\mathbb{R}^{n+1}$ (e.g., the origin)
$E$ is the fixed, expected rate of expansion of $(G_r^{\star})_{r\in\mathbb{N}}$ w.r.t a reference point $C$: e.g., $E=1$
$\mathcal{E}(C,G_{r}^{\star})$ is the actual rate [4, $\S$2] of expansion of $(G_r^{\star})_{r\in\mathbb{N}}$ w.r.t a reference point $C$

$\S$4.2. Leading Question.

Does there exist an unique choice function, which chooses an unique set $\mathcal{B}\subset\mathbf{B}(\mathbb{R}^{A})$ and $\mathscr{B}\subset\mathfrak{B}({\mathbb{R}}^{n})$($\S$3.1), where for all $r\in\mathbb{N}$, $f_r^{\star}\in\mathcal{B}$ and $A_r^{\star}\in\mathscr{B}$, such that:

$(f_r^{\star},A_r^{\star})\to(f,A)$($\S$2.1)
For all $v\in\mathbb{N}$, where for all $f_{v}^{\star\star}\in\mathbf{B}(\mathbb{R}^{A})\setminus\mathcal{B}$ and $A_{v}^{\star\star}\in\mathfrak{B}({\mathbb{R}}^{n})\setminus\mathscr{B}$($\S$3.1), assuming $(f_v^{\star\star},A_{v}^{\star\star})\to(f,A)$($\S$2.1), the "measure" [5, $\S$3] of $(G_{r}^{\star})_{r\in\mathbb{N}}=(\text{graph}(f_r^{\star}))_{r\in\mathbb{N}}$($\S$4.1) must increase at a rate linear or superlinear to that of $(G_{v}^{\star\star})_{v\in\mathbb{N}}=(\text{graph}(f_v^{\star\star}))_{v\in\mathbb{N}}$($\S$4.1)
$\mathbb{E}[f_r^{\star}]$ is unique and finite ($\S$2.2)
For some $f_r^{\star}\in\mathcal{B}$ and $A_r^{\star}\in\mathscr{B}$ satisfying 1., 2. and 3., when $f$ is unbounded (i.e, skip 4. when $f$ is bounded), for all sets $\mathcal{B}^{\prime}\subset\mathbf{B}(\mathbb{R}^{A})$ and $\mathscr{B}^{\prime}\subset\mathfrak{B}({\mathbb{R}}^{n})$, where $\star \mapsto \star\star\star$, $r\mapsto s$, $\mathcal{B}\mapsto \mathcal{B}^{\prime}$, and $\mathscr{B}\mapsto\mathscr{B}^{\prime}$ in 1., 2. and 3., s.t.$\,\neg\square(\mathbb{E}[f_r^{\star}]=\mathbb{E}[f_{s}^{\star\star\star}])$($\S$2.2,$\S$4.1), note for all $s\in\mathbb{N}$, where $f_s^{\star\star\star}\in\mathcal{B}^{\prime}$ and $A_{s}^{\star\star\star}\in\mathscr{B}^{\prime}$ satisfy 1., 2. and 3.:
- If the absolute value is $||\cdot||$ and the $(n+1)$-th coordinate of $C$($\S$3.1) is $x_{n+1}$, then $||\mathbb{E}[f_r^{\star}]-x_{n+1}||\le ||\mathbb{E}[f_{s}^{\star\star\star}]-x_{n+1}||$($\S$2.2,$\S$4.1)
- The "rate of divergence" of $||\mathcal{E}(C,G_{r}^{\star})-E||$($\S$4.1) is less than or equal to the "rate of divergence" of $||\mathcal{E}(C,G_{s}^{\star\star\star})-E||$($\S$4.1). In rigorous terms, when $r\in\mathbb{N}$, then for any linear ${s}_{1}:\mathbb{N}\to\mathbb{N}$, where $s=s_1(r)$ and the Big-O notation is $\mathcal{O}$, there exists a function $K:\mathbb{R}\to\mathbb{R}$, where the absolute value is $||\cdot||$ and ($\S$4.1):\begin{align*} ||\mathcal{E}(C,G_{r}^{\star})-E||=& \mathcal{O}(K(||\mathcal{E}(C,G_{s}^{\star\star\star})-E||))\\= & \mathcal{O}(K(||\mathcal{E}(C,G_{s_{1}(r)}^{\star\star\star})-E||)) \end{align*}such that: $$0\le\lim\limits_{x\to+\infty}K(x)/x<+\infty$$
When set $F\subset \mathbb{R}^{A}$ is the set of all $f\in \mathbb{R}^{A}$, where some choice function chooses collections $\mathcal{B}\subset\mathbf{B}(\mathbb{R}^{A})$($\S$3.1) and $\mathscr{B}\subset\mathfrak{B}({\mathbb{R}}^{n})$($\S$3.1), such that $f_r^{\star}\in\mathcal{B}$ and $A_r^{\star}\in\mathscr{B}$ satisfy 1., 2., 3. and 4., then $F$ should be:
- a prevalent ($\S$2.3) subset of $\mathbb{R}^{A}$
- If not prevalent ($\S$2.3) then neither a prevalent nor shy ($\S$2.3) subset of $\mathbb{R}^{A}$
Out of all choice functions which satisfy 1., 2., 3., 4., and 5., we choose the one with the simplest form, meaning for each choice function fully expanded, we take the one with the fewest variable/numbers?