Finding a non-trivial, rigorous definition of a "measure zero" and "full measure" subset of a set of function spaces which satisfies the following?

Background: I decided to ask a new question based on the advice of the user here and here.

Suppose $n\in\mathbb{N}$ and $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ is a function, where $A$ and $f$ are Borel.

In Section $\S$1, I explain what I want. In Section $\S$2, I explain the definitions. Then, in Section $\S$3, I list an attempt.

---

$\S$1. What we want

Question: Assuming $\mathrm{S}(n)=\left\{\mathfrak{F}_1:\mathfrak{F}_1\in\mathbb{R}^\mathrm{K},\mathrm{K}\subseteq\mathbb{R}^{n} \text{ and $\mathfrak{F}_1$ are Borel}\right\}$ is the set of all $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ (i.e., $A$ and $f$ are Borel), what is an example of a non-trivial, rigorous definition of a "full measure" and "measure zero" subset of $\mathrm{S}(n)$ which can be used to prove the following? For each $n\in\mathbb{N}$:

1. If $F\subset\mathrm{S}(n)$ is the set of all $\dot{\mathbf{f}}\in\mathrm{S}(n)$, where $\mathcal{D}:=\mathrm{dom}(\dot{\mathbf{f}})$ and $m_{\dot{\mathbf{f}}}(\mathcal{D};n)$ (Definition 2.1) is finite, then $F$ is a "measure zero" subset of $\mathrm{S}(n)$.

This means the set of all $f$ with a finite mean is a "measure zero" subset of $S(n)$. In other words, "almost no" $f$ has a finite mean.

2. If $F\subset\mathrm{S}(n)$ is the set of all $\dot{\mathbf{f}}\in\mathrm{S}(n)$, where $\mathcal{D}:=\mathrm{dom}(\dot{\mathbf{f}})$ and there exists $(f_r,A_r)\to(\dot{\mathbf{f}},\mathcal{D})$ (Definition 2.2) such that $m_{\dot{\mathbf{f}},\mathcal{D}}((f_r),(A_r);n)$ (Definition 2.3) is finite, then $F$ is a "full measure" subset of $\mathrm{S}(n)$.

This means the set of all $f$, where there exists a sequence of bounded functions converging to $f$ with a finite mean (Definition 2.3), forms a "full measure" subset of $S(n)$. In other words, "almost all" $f$ has at least one sequence of bounded functions converging to $f$ with a finite mean (Definition 2.3).

3. If $F\subset\mathrm{S}(n)$ is the set of all $\dot{\mathbf{f}}\in\mathrm{S}(n)$, where $\mathcal{D}:=\mathrm{dom}(\dot{\mathbf{f}})$ and there exists $(f_r,A_r),(g_v,B_v)\to(\dot{\mathbf{f}},\mathcal{D})$ (Definition 2.2) such that $m_{\dot{\mathbf{f}},\mathcal{D}}((f_r),(A_r);n)\neq m_{\dot{\mathbf{f}},\mathcal{D}}((g_v),(B_v);n)$ (Definition 2.3), then $F$ is a "full measure" subset of $\mathrm{S}(n)$.

This means the set of all $f$, where two or more sequences of bounded functions converging to $f$ have non-equivelant means (Definition 2.3), is a "full measure" subset of $S(n)$. In other terms, the "new mean" (Definition 2.3) of "almost all" $f$ are non-unique.

---

$\S$2. Definitions

Definition $\S$2.1 (Extended Mean of $f$ w.r.t. The Hausdorff Measure in the Hausdorff dimension of its domain)

Suppose:

• $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension
• $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.
• the integral of $f$ is defined, w.r.t. the Hausdorff measure in its dimension
• $(A_r)_{r\in\mathbb{N}}$ is a sequence of bounded sets
• $A$ is a Borel subset of $\mathbb{R}^n$
• $\mathbf{B}(A)$ is the set of all sequences of bounded sets with set-theoretic limit $A$

The mean of $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ (i.e., $f$ is Borel), w.r.t. the Hausdorff measure in its dimension, is $m_{f}(A;n)$ when it exists:

$$\small{\begin{align}& \forall((A_r)_{r\in\mathbb{N}}\!\in\!\mathbf{B}(A))\exists\!\:!(m_{f}(A;n)\in\mathbb{R})\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\\ &\left(r\ge N\Rightarrow\left|\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A_r)}(A_r)}\int_{A_r}f\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A_r)}-m_{f}(A;n)\right|<\epsilon\right) \end{align}}$$

The extended mean of $f$ extends the mean, w.r.t. the Hausodorff measure in its dimension $$\frac{1}{{\mathcal{H}}^{\dim_{\text{H}}(A)}(A)}\int_{A} f {} d{\mathcal{H}}^{\dim_{\text{H}}(A)}$$ for certain cases of unbounded $A$. In simpler terms, the extended mean of $f$ exists, when the mean on all sequences of bounded functions $(A_r)_{r\in\mathbb{N}}$ with set-theoretic limit $A$ are unique and finite.

Definition $\S$2.2 (Sequence of Bounded Functions With Different Domains Converging to $f$)

The sequence of functions $(f_r)_{r\in\mathbb{N}}$, where $(A_r)_{r\in\mathbb{N}}$ is a sequence of bounded sets and $f_{r}:A_r\to\mathbb{R}$ is a bounded function, converges to $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ (i.e., $A$ and $f$ are Borel) when:

For any $(x_1,\cdots,x_n)\in A$, there exists an $n$-dimensional sequence $(s_1(r),\cdots,s_n(r))\in A_r$ such that $(s_1(r),\cdots,s_n(r))\to (x_1,\cdots,x_n)$ and $f_r(s_1(r),\cdots,s_n(r))\to f(x_1,\cdots,x_n)$.

This is equivalent to:

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

Definition $\S$2.3 (Mean of a Sequence of Bounded Functions With Different Domains Converging to $f$)

Suppose:

• $(f_r,A_r)\to(f,A)$ (Definition 2.2)
• $|\cdot|$ is the absolute value
• $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension
• $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra
• the integral of $f$ is defined, w.r.t the Hausdorff measure in its dimension

The mean of $(f_r)_{r\in\mathbb{N}}$ is a real number $m_{f,A}((f_r),(A_r);n)$, when the following is true:

$$\scriptsize{\begin{align}&\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{{\mathcal{H}}^{\text{dim}_{\text{H}}(A_{r})}(A_{r})}\int_{A_{r}}f_{r}\, d{\mathcal{H}}^{\text{dim}_{\text{H}}(A_{r})}-m_{f,A}((f_r),(A_r);n)\right|< \epsilon\right) \end{align}}$$

when no such $m_{f,A}((f_r),(A_r);n)$ exists, $m_{f,A}((f_r),(A_r);n)$ is infinite or undefined. (If $m_{f,A}((f_r),(A_r);n)$ is infinite or undefined, replace $\mathcal{H}^{\text{dim}_{\text{H}}(A_{r})}$ with the generalized Hausdorff measure ${\mathscr{H}^{\phi_{h,g}^{\mu}(q,t)}}$.)

---

$\S$3. Attempt

According to this user, we want to choose a marginal distribution. I assume I want a uniform marginal distribution; however, I know little about statistics.

According to the user:

(Another way to think about it: In general we think of a probability distribution as an infinite-information analogue, so to speak, of an optimal encoding/Huffman tree. So what is the encoding, here?)

I want a more analytical definition such as prevelant or shy sets. However, $S(n)$ is not a vector space, so we cannot use either of those definitions.

Since I know little about advanced mathematics and statistics, see this paper for more context.