Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff dimension of set $A$.
## 1. Motivation ##
If $n\in\mathbb{N}$, where set $A\subseteq\mathbb{R}^{n}$ and the expected value of $f:A\to\mathbb{R}$ is
$$\mathbb{E}[f]=\frac{1}{{H}^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f \, dH^{\text{dim}_{\text{H}}(A)}$$
we can see there are cases where $\mathbb{E}[f]$ is undefined or infinite (e.g. ${H}^{\text{dim}_{\text{H}}(A)}(A)$ is zero, $+\infty$ or $f$ is unbounded).
One solution to getting a finite expected value is
1. Defining a [dimension function][1]; i.e., $h:[0,+\infty)\to[0,+\infty]$, that's monotonically increasing, strictly positive and right continuous such that, when $R$ denotes the radius of a ball in a covering for the definition of the Hausdorff Measure, we replace $R^{\text{dim}_{\text{H}}(A)}$ with $h(R)$, so $H^{h}(A)$ is positive and finite.
Note, however, not all $A$ has a dimension function which leads to:
2. If $A$ is fractal but has no gauge function, we could use the [following][2]; however, when $A$ is non-fractal (e.g. countably infinite), we have that $\mathbb{E}[f]$ is undefined.
## 2. Solution for Bounded $f$ ##
Thus, we want to define a sequence of sets $(F_r)_{r\in\mathbb{N}}$ where if:
- $\liminf\limits_{r\to\infty} F_r=\bigcup\limits_{r\ge 1}\bigcap\limits_{q\ge r}F_q$
- $\limsup\limits_{r\to\infty} F_r=\bigcap\limits_{r\ge 1}\bigcup\limits_{q\ge r}F_q$
we want:
1. $\liminf\limits_{r\to\infty} F_r=\limsup\limits_{r\to\infty} F_r=A$ which I'll desribe as $F_r\overset{r\in\mathbb{N}}{\rightrightarrows} A$ or "a sequence of sets converging to $A$"
2. For all $r\in\mathbb{N}$,
- $0<{H}^{\dim_{\text{H}}(A)}(F_r)<+\infty$
- $0<{H}^{h}(F_r)<+\infty$
If criteria 1. and 2. are satisfied, the *new* finite expected value of $f$ w.r.t $F_r$ or $\mathbb{E}^{*}\left[f,F_r\right]$ is:
\begin{align}
& \forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{{H}^{h}\left(F_r\right)}\int_{F_r}f\, d{H}^{h}-\mathbb{E}^{*}[f,F_r]\right|< \epsilon\right) \\
\end{align}
when $f$ is bounded.
## 3. Equivalent and Non-Equivalent Sequence of Sets ##
> Note we can determine whether $\left(F_r\right)_{r\in\mathbb{N}}$ is
> **equivelant** to another sequence of sets converging to $A$ (e.g. $(F_j^{\prime})_{j\in\mathbb{N}}$), if for all
> $f\in\mathbb{R}^{A}$ we have $\mathbb{E}^{*}[f,F_r]$ or
> $\mathbb{E}^{*}[f,F_j^{\prime}]$ exists such that:
>
> $$\mathbb{E}^{*}[f,F_r]=\mathbb{E}^{*}[f,F_j^{\prime}]$$
>
> If however, there exists an $f\in\mathbb{R}^{A}$ where:
>
> $$\mathbb{E}^{*}[f,F_r]\neq\mathbb{E}^{*}[f,F_j^{\prime}]$$
>
> then the sequence of sets is **non-equivalent**.
## 4. Solution for Unbounded $f$ ##
The problem is when $f$ is unbounded, no matter what $F_r$ is chosen, we might have $\mathbb{E}^{*}[f,F_r]$ is undefined. For instance, if $f$ is continuous on set $A=\mathbb{R}\setminus\left\{0\right\}$ where $f(x)=1/x$, then the expected value of $f$ would still be undefined.
To solve this, consider the following:
If the image of $f$ under $A$ is $f[A]:=\left\{f(x):x\in A\right\}$ and the pre-image under $f$ of $F_{r}$ is $f^{-1}\left[F_r\right]:=\left\{x\in A: f(x)\in F_r\right\}$, we want:
1. $F_{r}^{\dagger}\overset{r\in\mathbb{N}}{\rightrightarrows} A\times f[A]$
2. For all $r\in\mathbb{N}$,
- $0<{H}^{\dim_{\text{H}}(A\times f[A])}(F_r^{\dagger})<+\infty$
- For exact dimension function $h$ of set $A\times f[A]$, $0<{H}^{h}(F_r^{\dagger})<+\infty$
where $\dagger$ symbolizes a sequence of sets that should converge to $A\times f[A]$, e.g. $F_{j}^{\dagger \dagger}$ or $F_{k}^{\dagger \dagger \dagger}$ must converge to $A\times f[A]$
**4.1. Generalized Expected Value:** Thus, the *generalized expected value* of $f$ (with respect to $F_r$) is $\mathbb{E}^{**}[f,F_{r}^{\dagger}]$, when it exists where:
\begin{align}
& \small{\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{{H}^{h}\left(f^{-1}[F_r^{\dagger}]\right)}\int_{f^{-1}[F_r^{\dagger}]}f\, d{H}^{h}-\mathbb{E}^{**}[f,F_r^{\dagger}]\right|< \epsilon\right)} \\
\end{align}
## 5. Choosing a specific sequence of sets ##
The problem is "most" unbounded $f$ have multiple $F_r^{\dagger}$ that give different $\mathbb{E}^{**}[f,F_{r}^{\dagger}]$. Therefore, we should choose a set of equivalent $F_{r}^{\dagger}$ using a choice function.
Here is what I considered:
**5.1 Question**
Does there exist a choice function which chooses a unique set (of equivalent sequences of sets) that converge to $A\times f[A]$, such that:
1. The sequence of sets converge to $A\times f[A]$ at a *linear* or *super-linear* rate compared to the rate non-equivelant sequences of sets converge to $A\times f[A]$
2. The *generalized expected value* of $f$ w.r.t the chosen (and equivalent) sequences of sets is finite.
3. The choice function chooses a unique set of equivalent sequences of sets which satisfy 1. and 2., for all $f\in Q$, where set $Q\subseteq\mathbb{R}^{A}$ with $Q$ [prevalent][3] in $\mathbb{R}^{A}$. (This means the choice function chooses a unique and equivalent sequence of sets, which satisfy 1. and 2. for "almost all" $f\in\mathbb{R}^{A}$).
4. Out of all the choice functions which satisfy 1., 2. and 3., we choose the one with the *simplest form*, meaning for each choice function fully expanded, we take the one with the fewest variables/numbers (excluding those with quantifiers).
**5.2 Notes on Question**
If the solution is *extraneous*, what other criteria should be included to get a unique choice function? (Note if the solution is *always extraneous*, we want to replace the phrase “equivelant sequences of sets” with the following: ”the set of all sequences of sets, such that the generalized expected values of $f$ w.r.t each sequence are the same”.)
See my attempt in the answer (once posted) for what a solution can look like.
[1]: https://en.wikipedia.org/wiki/Dimension_function#:~:text=In%20mathematics%2C%20the%20notion%20of,of%20s%2Ddimensional%20Hausdorff%20measure.
[2]: https://arxiv.org/pdf/math/9405217.pdf
[3]: https://www.ams.org/journals/bull/2005-42-03/S0273-0979-05-01060-8/S0273-0979-05-01060-8.pdf