Finding an Unique and “Natural” Extension of the Expected Value That Takes a Finite Value for All Functions in Prevalent Subset of the Set of All Functions

1. Introduction

According to an article in Quanta Magazine [2] Wood writes, "No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe." The cited paper [3] presents a constructive approach to Wood’s statement using filters over families of finite set; however, the average in the approach is not unique: the method determines the average value of functions with a range that lies in any algebraic structure for which the finite averages make sense. In this paper, we will explore a more constructive approach where the average is unique, finite, and "natural" (defined in $\S$3.3 & $\S$3.4) for a non-shy subset [4] of the set of measurable functions. (Note the functions must be measurable for application purposes).

We begin by describing "the infinite objects" which cover "an infinite expanse of space" as unbounded functions, since these functions are approachable from a mathematical standpoint. Moreover, if we define $n\in \mathbb{N}$, where set $A\subseteq {\mathbb{R}}^n$ and function $f:A\to \mathbb{R}$; suppose a prevalent subset of a function-space means "almost all" functions are in that space, a shy subset of a function-space means "almost no" functions are in that space and $B^{*}$ is the set of all Borel measurable functions in ${\mathbb{R}}^A$. We then get the set of unbounded f where the expected value is infinite or undefined, forms a non-shy (i.e., prevalent nor shy or prevelant) subset of $B^{*}$. Furthermore, the set of all f with a finite expected value forms only a shy subset of $B^{*}$, meaning only a "negligible" amount of measurable functions have finite expected values.

Therefore, when defining prevalent and shy sets using mathematics in $\S$2.1; we’ll define four attempts to answer the thesis 1 of the first paragraph of $\S$2.2. Note neither attempts give complete answers: they extend the Hausdorff measure of A to be positive and finite for "most" subsets of ${\mathbb{R}}^n$ but don’t guarantee that unbounded functions in a non-shy subset of measurable functions have finite expected values. Infact, the expected value from all attempts might be positive and finite for only a shy subset of $B^{*}$.

Hence, we define a sequence of sets called ★-sequence of sets (def. 8) whose properties allow for finite expected values for a non-shy subset of $M^{*}$. Note these ★-sequences of sets converge to the graph of f i.e. $\left\{(x,f(x\left)\right):x\in A\right\}$ rather than A; otherwise, the generalized expected value of f w.r.t to a ★-sequence (def. 9) cannot, in general, be finite for unbounded functions. Moreover, since there are functions with multiple ★-sequences of sets, where generalized expected values of f w.r.t each ★-sequence are different and non-unique—we must have a choice function which chooses a unique set of equivalent ★-sequences with the same, unique expected value.

For defining the choice function, we ask a question in $\S$3.4 where with previous sections; we define equivalent & non-equivalent ★-sequences of sets for $\S$3.1, as well as "natural" expected values for $\S$3.3. We attempt to answer the question in $\S$3.4 by redefining linear/super-linear convergence (def. 12) in terms of entropy, samples and "pathways" where the samples are derived by taking a point from each partition of a ★-sequence of sets, such the partitions have equal Hausdorff measure. Since all samples have finite points; we take a "pathway" of line segments between points in the sample, where in def. 15 we exclude segments with "extra-large" proportion to a chosen measure for each partition of ★-sequence of sets). The procedure is similar to the ones used in computers to graph functions [5]. We also take the length of each of the line segments in the "pathway", multiplying all lengths by a constant so they add up to one (i.e. a discrete probability distribution). We take the supremum of the Entropy of the distribution [6] w.r.t all "pathways" to redefine def. 12 as def. 16 such redefined definition is used to create a choice function in $\S$5.1.

2. Preliminary Definitons/Motivation

Other than integration with filters [3], there is no constructive approach to finding a meaningful average of functions covering an infinite expanse of space; however, there are two constructive approaches to making the average unique, finite, and "natural". Before beginning, consider the following mathematical definitions:

2.1. Preliminary Definitions

Let X be a completely metrizable topological vector space.

Definition 1 

(Prevalent Subset of X). A Borel set $E\subset X$ is said to be prevalent if there exists a Borel measure μ on X such that:

• $0<\mu \left(C\right)<\infty$ for some compact subset C of X, and
• the set $E+x$ has full μ-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. Also note:

Definition 2 

(Shy Subset of X). The complement of a prevalent set is called a shy set.

such that we define:

Definition 3 

(Non-Shy Subset of X). A subset of X that is prevalent or neither prevalent nor shy.

Furthermore, suppose we define:

Definition 4 

(Hausdorff Measure). Let $(V,d)$ be a metric space, $\alpha \in [0,\infty )$. For every $C\in V$, define the diameter of C as:

$$\mathit{diam}\left(C\right):=sup\left\{d(x,y):x,y\in C\right\},\mathit{diam}(\varnothing ):=0$$

We define:

$$H_{\delta }^{\alpha }\left(E\right)=inf\left\{\sum _{i=1}^{\infty }{\left(\mathit{diam}\left(C_i\right)\right)}^{\alpha }:\mathit{diam}\left(C_i\right)\le \delta ,E\subseteq \bigcup _{i=1}^{\infty }{C}_i\right\}.$$

(2.1.1)

The Hausdorff Outer Measure is defined by

$$H^{\alpha }\left(E\right)=\underset{\delta >0}{sup}{H}_{\delta }^{\alpha }\left(E\right)=\underset{\delta \to 0}{lim}{H}_{\delta }^{\alpha }\left(E\right)$$

If $i\in \mathbb{N}$ and $\delta \in \mathbb{R}$ such that $\delta >0$, where the Euler’s Gamma function is Γ and constant ${\mathcal{N}}_{\alpha }$ is:

$${\mathcal{N}}_{\alpha }=\frac{{\pi }^{\alpha /2}}{2\mathsf{\Gamma }\left(\frac{\alpha }{2}+1\right)}$$

(2.1.2)

when $\alpha \in \mathbb{N}$ and E is a Borel set we have that

$$L^{\alpha }\left(E\right)=\frac{1}{2}{\mathcal{N}}_{\alpha }{H}^{\alpha }\left(E\right)$$

(2.1.3)

such that $H^{\alpha }\left(E\right)$ is related to the α-dimensional Lebesgue Measure.

Definition 5 

(Hausdorff Dimension). The Hausdorff Dimension of E is defined by ${\dim }_{\mathrm{H}}\left(E\right)$ where:

$${\mathcal{H}}^{\alpha }\left(E\right)=\left\{\begin{array}{cc}\infty \hfill & \mathrm{if}0\le \alpha <{\mathit{dim}}_{\mathrm{H}}\left(E\right)\hfill \\ 0\hfill & \mathrm{if}{\mathit{dim}}_{\mathrm{H}}\left(E\right)<\alpha <\infty \hfill \end{array}\right.$$

(2.1.4)

Therefore, we can use definitions 1, 2, 4 to prove or disprove:

Theorem 1.

The set of unbounded functions forms a prevalent subset of the set of all measurable functions.

[Notes on Theorem 1]

By measurable function, we mean the pre-image of any subset of $\mathbb{R}$ (under a measurable function) is in the sigma-algebra of the Hausdorff measure. (Note function f on set A is unbounded when there is no $I\ge 0$ such that for all $x\in A$):

$$\left|f\left(x\right)\right|\le I$$

however, we’re unsure if theorem 1 is correct. Despite this, we could prove or disprove theorem 1 using the paper on prevalence in [4].

We, therefore, define the expected value w.r.t the Hausdorff measure to be the following:

Definition 6 

(Expected Value of f).If $n\in \mathbb{N}$, where set $A\subseteq {\mathbb{R}}^n$, the expected value of function $f:A\to \mathbb{R}$ (using def. 4 and 5) is

$$\mathbb{E}\left[f\right]=\frac{1}{H^{{\mathit{dim}}_{\mathrm{H}}\left(A\right)}\left(A\right)}{\int }_{A}fd{H}^{{\mathit{dim}}_{\mathrm{H}}\left(A\right)}$$

where we can see there are cases where $\mathbb{E}\left[f\right]$ is undefined or infinite (e.g. $H^{{\mathit{dim}}_{\mathrm{H}}\left(A\right)}\left(A\right)$ is zero, $+\infty$ or f is unbounded). In this case, if topological vector space X is ${\mathbb{R}}^A$ (see $\S$) where we define $B^{*}$ such that:

Definition 7 

(The set of all measurable functions). $B^{*}$ is the set of all Borel measurable functions in ${\mathbb{R}}^A$.

Then, we must prove:

Theorem 2.

The expected value $\mathbb{E}\left[f\right]$ is finite for all f in only a shy subset of $B^{*}$.

[Note on Theorem 2]

We’re not sure how to prove theorem 2; however, we refer to an answer from @Mathe at the last page of this citation [7],

"We can follow the argument presented in example 3.6 of [4]:

Because a function can always be represented as $f=f^{+}-f^{-}$ we only consider whether positive functions have a mean value. We consider the case of a set A with finite positive measure. In this context having a mean means having a finite integral, and not being integrable means having an infinite integral.

Take $X:=L^0\left(A\right)$ (measurable functions over A) let P denote the one-dimensional subspace of $L^0\left(A\right)$ consisting of constant functions (assuming the Hausdorff measure on A) and let $F:=L^0\left(A\right)\setminus L^1\left(A\right)$ (measurable functions over A with no finite integral)

If ${\lambda }_P$ denotes the Lebesgue measure over P, for any fixed $f\in F$

$${\lambda }_P\left(\left\{\beta \in \mathbb{R}:{\int }_A(f+\beta )\mu <\infty \right\}\right)=0$$

Meaning P is a 1-dimensional probe of F, so F is a 1-prevalent set. (In other terms, the set of measurable functions over A with no finite integral or mean, forms a prevalent subset of the set of all measurable functions in ${\mathbb{R}}^A$. Therefore, using def. 2, the set of measurable functions with a finite integral or mean forms a shy subset of all Borel measurable functions in ${\mathbb{R}}^A$.)

2.2. Extended Expected Values

Four solutions to getting a finite expected value for "larger" subset of ${\mathbb{R}}^A$ is:

• Defining a dimension function; 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 $R^{{\dim }_{\mathrm{H}}\left(A\right)}$ with $h\left(R\right)$ so $H^h\left(A\right)$: the h-Hausdorff measure, is positive and finite. This leads to the extended expected value ${\mathbb{E}}^{★}\left[f\right]$, where:

  $${\mathbb{E}}^{★}\left[f\right]=\frac{1}{H^h\left(A\right)}{\int }_{A}fd{H}^h$$
  Note, however, not all A has dimension function h which leads to:
• If A is fractal but has no gauge function, we could use this paper [8] which is an extension of the Lebesgue density theorem and this paper [9] which is an extension of the Hausdorff measure using Hyperbolic Cantor sets. Note, however, when A is non-fractal (e.g. countably infinite) or f is unbounded, there is a possibility that the expected value is infinite or undefined. Hence,
• In the case f is unbounded and fractal, we could use [10], which applies a Henstock-Kurzweil type integral (i.e., $\mu$-HK integral) on a measure Metric Space. This coincides with unbounded functions with finite improper Riemman integrals, including bounded functions with finite Lebesgue integrals, bounded function with finite integrals w.r.t the Hausdorff measure, or function with finite Henstock-Kurzweil integrals.

2.3. Examples

If $n\in \mathbb{N}$, set $A\subseteq {\mathbb{R}}^n$ and function $f:A\to \mathbb{R}$, we want to apply the definitions of the next section for the following examples:

(a)

$A=\mathbb{R}\setminus \left\{0\right\}$ and $f\left(x\right)=1/x$. This function is unbounded and has an undefined expected value since the average of $1/x$, using the improper Riemann integral on $\mathbb{R}\setminus \left\{0\right\}$:

$$\begin{array}{cc}\hfill & \underset{(x_1,x_2,x_3,x_4)\to (-\infty ,0^{-},0^{+},+\infty )}{lim}\frac{1}{(x_4-x_3)+(x_2-x_1)}\left({\int }_{x_1}^{x_2}\frac{1}{x}dx+{\int }_{x_3}^{x_4}\frac{1}{x}dx\right)=\hfill \end{array}$$

(2.3.1)

$$\begin{array}{cc}\hfill & \underset{(x_1,x_2,x_3,x_4)\to (-\infty ,0^{-},0^{+},+\infty )}{lim}\frac{1}{(x_4-x_3)+(x_2-x_1)}\left(ln\left(\right|x\left|\right)+C{|}_{x_1}^{x_2}+ln\left(\right|x\left|\right)+C{|}_{x_3}^{x_4}\right)=\hfill \end{array}$$

(2.3.2)

$$\begin{array}{cc}\hfill & \underset{(x_1,x_2,x_3,x_4)\to (-\infty ,0^{-},0^{+},+\infty )}{lim}\frac{1}{(x_4-x_3)+(x_2-x_1)}\left(ln\left(\right|x_2\left|\right)-ln\left(\right|x_1\left|\right)+ln\left(\right|x_4\left|\right)-ln\left(\right|x_3\left|\right)\right)\hfill \end{array}$$

(2.3.3)

is $+\infty$ (when $x_2=1/x_1$, $x_3=1/x_4$, and $x_1=exp\left(x_4^2\right)$) or $-\infty$ (when $x_2=1/x_1$, $x_3=1/x_4$, and $x_4=-exp\left(x_1^2\right)$), making the average undefined.

(b)

$A=\mathbb{Q}$, gcd is the greatest common divisor, and $f_1,f_2:\mathbb{R}\to \mathbb{R}$ where:

$$f\left(x\right)=\left\{\begin{array}{cc}{f}_1\left(x\right)\hfill & x\in A_1:=\left\{r/q:r\in \mathrm{odd}\mathbb{Z},q\in \mathrm{even}\mathbb{Z},q\ne 0,gcd(r,q)=1\right\}\hfill \\ f_2\left(x\right)\hfill & x\in A_2:=\left\{r_1/\left(q_1\right):r_1\in \mathbb{Z},q_1\in \mathrm{odd}\mathbb{Z},gcd(r_1,q_1)=1\right\}\hfill \end{array}\right.$$

(2.3.4)

For instance, point $(1/4,1)$ is a point in the graph of f (since $1/4\in \mathbb{Q}$ and $1/4\in A_1$, making $f(1/4)=f_1(1/4)$). Also, point $(1/3,0)$ is a point in the graph of f (since $1/3\in \mathbb{Q}$ and $1/3\in A_2$, making $f(1/3)=f_2(1/3)$); however, point $(\sqrt{2},1)$ is not in the graph of f (since $\sqrt{2}\notin \mathbb{Q}$).

Note the function in eq. 2.3.4 is bounded; however, the expected value & extensions are undefined. (Using def. 6, we know ${\dim }_{\mathrm{H}}\left(A\right)=0$ but $H^{{\dim }_H\left(A\right)}\left(A\right)=+\infty$, which makes $\mathbb{E}\left[f\right]$:

$$\mathbb{E}\left[f\right]=\frac{1}{H^{{\dim }_{\mathrm{H}}\left(A\right)}\left(A\right)}{\int }_{A}fd{H}^{{\dim }_{\mathrm{H}}\left(A\right)}$$

undefined by division of $+\infty$.) Further, we assume using $\S$2.2, crit. 1, there is no (exact) dimension function of A nor could A be "fractal" enough for extensions of the Lebesgue Density Theorem [8], extensions of the Hausdorff measure using Hyperbolic Cantor Sets [9], or extension of the Henstock-Kurzweil integral on the Metric Space [10]. Lastly, as stated in $\S$2.2, crit. ??, the conditional expected value of the function in eq. 8 varies with the "condition" chosen (see $\S$??, ex. ??).

3. Attempt to Answer Thesis

Suppose h is the dimension function, $H^h$ is the h-Hausdorff measure ($\S$2.2, crit. 1) and $M^{*}$ is the set of all measurable functions in ${\mathbb{R}}^A$.

Definition 8 

(★-Sequence of Sets). If we define a sequence of sets ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$, where h is the dimension function, then when:

(a)

The set theoretic limit of ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ is the graph of f (i.e., ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ converges to the graph of f) where

$$\begin{array}{c}\underset{r\to \infty }{lim\; sup}{F}_r^{★}=\bigcap _{r\ge 1}\bigcup _{q\ge r}{F}_q^{★}\\ \underset{r\to \infty }{lim\; inf}{F}_r^{★}=\bigcup _{r\ge 1}\bigcap _{q\ge r}{F}_q^{★}\end{array}$$

with the graph of f as:

$$\left\{(x,f(x\left)\right):x\in A\right\}$$

the set-theoretic limit should be:

$$\underset{r\to \infty }{lim\; sup}{F}_r^{★}=\underset{r\to \infty }{lim\; inf}{F}_r^{★}=\left\{(x,f(x\left)\right):x\in A\right\}$$

(b)

For all $r\in \mathbb{N}$, where $H^h$ is the h-Hausdorff measure ($\S$2.2, crit. 1),

$$0<H^h\left(F_r^{★}\right)<+\infty$$

(c)

we define sequence of functions ${\left(f_r^{★}\right)}_{r\in \mathbb{N}}$ where $f_r^{★}:\mathit{dom}\left(F_r^{★}\right)\to \mathit{range}\left(F_r^{★}\right)$ such that$\left\{(x,f_r^{★}\left(x\right)):x\in \mathit{dom}\left(F_r^{★}\right)\right\}=F_r^{★}$

we have $\left(F_r^{★}\right)$ is a ★-sequence of sets or starred-sequence of sets.

Example 1.

One ★-sequence of sets of $f\left(x\right)=1/x$ on $\mathbb{R}\setminus \left\{0\right\}$ ($\S$2.3, crit. 3a) is:

$${\left(F_r^{★}\right)}_{r\in \mathbb{N}}={\left(\left\{(x,1/x):x\in [-r,-1/r]\cup [1/r,r]\right\}\right)}_{r\in \mathbb{N}}$$

Example 2.

Another example of a ★-sequence of sets of $f:\mathbb{Q}\to \mathbb{R}$ where:

$$f\left(x\right)=\left\{\begin{array}{cc}1\hfill & x\in A_1:=\left\{r/q:r\in \mathrm{odd}\mathbb{Z},q\in \mathrm{even}\mathbb{N},q\ne 0,gcd(r,q)=1\right\}\hfill \\ 0\hfill & x\in A_2:=\left\{r_1/\left(q_1\right):r_1\in \mathbb{Z},q_1\in \mathrm{odd}\mathbb{N},gcd(r_1,q_1)=1\right\}\hfill \end{array}\right.$$

(3.0.1)

using ($\S$2.3, crit. 3b) is the following:

$${\left(F_r^{★}\right)}_{r\in \mathbb{N}}={\left(\left(x,f\left(x\right)\right):x\in \left\{c/(r!):-r·r!\le c\le r·r!\right\}\right)}_{r\in \mathbb{N}}$$

(3.0.2)

another example is:

$${\left(F_r^{★}\right)}_{r\in \mathbb{N}}={\left(\left(x,f\left(x\right)\right):x\in \left\{c/d:d\le r,-d·r\le c\le d·r\right\}\right)}_{r\in \mathbb{N}}$$

(3.0.3)

Note this leads to a new extension of the expected value where when there’s at least one starred-sequence of sets where the extension is finite, the extension could be finite for all f in a prevalent subset of ${\mathbb{R}}^A$.

Definition 9.

(Generalized Expected Value). If ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ is a ★-sequence of sets (def. 8), the generalized expected value of f w.r.t ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ is $E^{**}[f,F_r^{★}]$ where:

$$\begin{array}{cc}\hfill & \forall (ϵ>0)\exists (N\in \mathbb{N})\forall (r\in \mathbb{N})\left(r\ge N\Rightarrow \left|\frac{1}{H^h\left(\mathrm{dom}\left(F_r^{★}\right)\right)}{\int }_{\mathrm{dom}\left(F_r^{★}\right)}{f}_r^{★}d{H}^h-{\mathbb{E}}^{**}[f,F_r^{★}]\right|<ϵ\right)\hfill \end{array}$$

(3.0.4)

Example 3.

Using example 1, we find that when ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}={\left(\left\{(x,1/x):x\in [-r,-1/r]\cup [1/r,r]\right\}\right)}_{r\in \mathbb{N}}$:

(a)

$\mathit{dom}\left(F_r^{★}\right)={\left([-r,-1/r]\cup [1/r,r]\right)}_{r\in \mathbb{N}}$

(b)

$f_r\left(x\right)=1/x$ for $x\in [-r,-1/r]\cup [1/r,r]$

and the generalized expected value is:

$$\begin{array}{cc}\hfill \underset{(x_1,x_2,x_3,x_4)\to (-\infty ,0^{-},0^{+},+\infty )}{lim}\frac{1}{(x_4-x_3)+(x_2-x_1)}& \left({\int }_{x_1}^{x_2}\frac{1}{x}dx+{\int }_{x_3}^{x_4}\frac{1}{x}dx\right)=\hfill \end{array}$$

(3.0.5)

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}\frac{1}{(r-1/r)+(-1/r-(-r\left)\right)}\left({\int }_{-r}^{-1/r}\frac{1}{x}dx+{\int }_{1/r}^r\frac{1}{x}dx\right)=\end{array}$$

(3.0.6)

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}\frac{1}{(r-1/r)+(-1/r+r)}& \left(ln\left(\right|x\left|\right)+C{|}_{-r}^{-1/r}+ln\left(\right|x\left|\right)+C{|}_{1/r}^r\right)=\hfill \end{array}$$

(3.0.7)

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}\frac{1}{(r-1/r)+(-1/r+r)}& \left(ln\left(\right|-r\left|\right)-ln\left(\right|-1/r\left|\right)+ln\left(\right|r\left|\right)-ln\left(\right|1/r\left|\right)\right)=\hfill \end{array}$$

(3.0.8)

$$\begin{array}{cc}\hfill \underset{r\to \infty }{lim}\frac{1}{2r-2/r}& ·4ln\left(r\right)=\hfill \end{array}$$

(3.0.9)

$$\begin{array}{cc}\hfill & 0\hfill \end{array}$$

(3.0.10)

We can see from example 1, the average was once undefined but now we’ve "chosen" a ★-sequence which gives a finite expected value.

3.1. Equivalent and Non-Equivalent ★-sequences of Sets

Next, we define set $V^{\prime }$, where we want the generalized expected value to exist for all $f\in V^{\prime }$ w.r.t at least one sequence (in a set of ★-sequences of sets) where

Definition 10.

(Non-Equivalent Starred-Sequences of Sets). All starred-sequences of sets are non-equivalent (in a set of ★-sequences of sets), if there exists an $f\in V^{\prime }$, where the generalized expected values of f w.r.t each starred-sequence of sets has two or more different values; e.g., defined vs undefined values.

Figure 1. Below $F_r^{★}$, $F_k^{★★}$, $F_z^{★★★}$ are non-equivalent starred sequences of sets, where $V^{\prime }$ is all circles and $E^{**}$ is the generalized expected value of f w.r.t either ★-sequence of sets (def. 8)

Figure 1. Below $F_r^{★}$, $F_k^{★★}$, $F_z^{★★★}$ are non-equivalent starred sequences of sets, where $V^{\prime }$ is all circles and $E^{**}$ is the generalized expected value of f w.r.t either ★-sequence of sets (def. 8)

Definition 11.

(Equivalent Starred-Sequences of Sets). All starred-sequences of sets are equivalent (in the set of ★-sequences of sets), if we get for all $f\in V^{\prime }$; the generalized expected value of f (def. 9) w.r.t each starred-sequence of sets has the same value.

Figure 2. Below $F_r^{★}$, $F_k^{★★}$, $F_z^{★★★}$ are equivalent starred sequences of sets, where $V^{\prime }$ is the entire circle and $E^{**}$ is the generalized expected value of f w.r.t either ★-sequence of sets (def. 8)

Figure 2. Below $F_r^{★}$, $F_k^{★★}$, $F_z^{★★★}$ are equivalent starred sequences of sets, where $V^{\prime }$ is the entire circle and $E^{**}$ is the generalized expected value of f w.r.t either ★-sequence of sets (def. 8)

However, proving that two or more starred-sequences of sets are non-equivalent or equivalent (using def. 11 or 10) is tedious. Therefore, we ask the following:

3.1.1. Question 1

Is there are a simpler definition of equivalent and non-equivalent ★-sequences of sets.

3.2. Motivation for Question

For all f in a prevalent subset of ${\mathbb{R}}^A$ (def. 1), we may choose a ★-sequence of sets ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ where the generalized expected value of f w.r.t least one starred-sequence is finite. However, consider the following problem:

Theorem 3.

The set of all f, where the generalized expected values of f w.r.t two or more non-equivalent ★-sequences of sets has different values, form a prevalent subset of all measurable functions in ${\mathbb{R}}^A$.

This means "almost all" measurable functions have several generalized expected values depending on the starred-sequence chosen. Therefore, we need to choose a unique ★-sequence of sets where the new extended expected value is also "natural" and unique:

3.3. Essential Definitions for a "Natural" Expected Value

Suppose ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ and ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$ are non-equivelant starred-sequences of sets (def. 8 & 10): we have the following is essential for a "natural" expected value.

Definition 12.

Definition 12 (Linear & Super-linear Convergence of a ★-Sequence of Sets To That Of Another ★-Sequence of Sets). If we define function $S:\mathbb{R}\to \mathbb{R}$, where $r,j\in \mathbb{N}$ such that:

$$H^h\left(F_r^{★}\right)=\mathcal{O}\left(S\left(H^h\left(F_j^{★★}\right)\right)\right)$$

where we have $\mathcal{O}$ as the Big-O notation and $0<\underset{x\to \infty }{lim}S\left(x\right)/x$, then ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ converges to the graph of f:

$\left\{(x,f(x\left)\right):x\in A\right\}$ at alinear or super-linearrate compared to that of ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$.

Now we may combine the previous definitions into a main question with an answer that solves the thesis 2.

3.4. Main Question

Does there exist a choice function that chooses a unique set (of equivalent ★-sequences of sets) such that:

(a)

The chosen starred-sequences of sets converge to $\left\{(x,f(x\left)\right):x\in A\right\}$ at a rate linear or super-linear (def. 12) to the rate non-equivalent ★-sequences of sets converge to $\left\{(x,f(x\left)\right):x\in A\right\}$

(b)

The generalized expected value (def. 9) of f w.r.t the chosen (and equivalent) starred-sequences of sets is finite.

(c)

The choice function chooses a unique set of equivalent ★-sequences of sets which satisfy (1) and (2), for all $f\in Q$ such that Q is a non-shy subset (def. 5) of $M^{*}$ (i.e., the set of all measurable functions).

(d)

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)?

[Notes On Question]

Note, the unique set of equivalent and chosen starred-sequences of sets is defined using notation $\sim {\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$, where ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ is a starred-sequence in $\sim {\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$. Therefore, after we define the choice function, the answer should be:

$$\begin{array}{cc}\hfill & \forall (ϵ>0)\exists (N\in \mathbb{N})\forall (k\in \mathbb{N})\left(k\ge N\Rightarrow \left|\frac{1}{H^h\left(\mathrm{dom}\left(F_k^{★★★}\right)\right)}{\int }_{\mathrm{dom}\left(F_k^{★★★}\right)}{f}_k^{★}d{H}^h-{\mathbb{E}}^{**}[f,F_k^{★★★}]\right|<ϵ\right)\hfill \end{array}$$

(3.4.1)

Also, consider the following: if the solution to the main question is extraneous, what other criteria can be included to get a unique choice function? (Note if the solution is always extraneous, we want to replace “equivelant starred-sequences of sets” with the following: ”the set of all ★-sequences of sets, where the generalized expected values of f w.r.t each starred-sequence is the same”.)

4. Solution To The Main Question Of Section 2.4

Suppose h is the dimension function, $H^h$ is the h-Hausdorff measure ($\S$2.2, crit. 1), and ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ is the starred-sequence of sets (def. 8). We will use an alternative approach to definition 12 or def. 16 so we can define a choice function which solves the main question. Read from the second sentence of the last paragraph of the intro of $\S$1 for a summary. Also, refer to sec. 3 and 4 of [1] for examples: (the cited paper uses sets instead of functions).

4.1. Preliminary Definitions

Definition 13.

(Uniform $\epsilon$ coverings of each term of a ★-sequence of sets). We define uniform ε coverings of each term of ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ as a group of pair-wise disjoint sets which cover $F_r^{★}$ (for some $r\in \mathbb{N}$), such when taking dimension function h of $F_r^{★}$, we want $H^h$ of each pair-wise disjoint set to have the same value $\epsilon \in \mathit{range}\left(H^h\right)$, where $\epsilon >0$ and the total sum of $H^h$ of the coverings is minimized. In shorter notation, if

• The element $t\in \mathbb{N}$
• The set $T\supset \mathbb{N}$ is arbitrary and uncountable.

and set Ω is defined as:

$$\mathsf{\Omega }=\left\{\begin{array}{cc}\left\{1,···,t\right\}\hfill & \mathit{if}\mathit{there}\mathit{are}t\mathit{ways}\mathit{of}\mathit{writing}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \\ \mathbb{N}\hfill & \mathit{if}\mathit{there}\mathit{are}\mathit{countably}\mathit{infinite}\mathit{ways}\mathit{of}\mathit{writing}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \\ T\hfill & \mathit{if}\mathit{there}\mathit{are}\mathit{uncountable}\mathit{ways}\mathit{of}\mathit{writing}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \end{array}\right.$$

(4.1.1)

then for every $\omega \in \Omega$, the set of uniform ε coverings is defined using $\mathcal{U}(\epsilon ,F_r^{★},\omega )$ where ω “enumerates" all possible uniform ε coverings of $F_r^{★}$ for every $r\in \mathbb{N}$.

Definition 14.

(Sample of the uniform $\epsilon$ coverings of each term of a ★-sequence of sets). The sample of uniform ε coverings of each term of ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ is the set of points where for every $\epsilon \in \mathit{range}\left(H^h\right)$ and $r\in \mathbb{N}$, we take a point from each pair-wise disjoint set in the uniform ε coverings of $F_r^{★}$ (def. 13). In shorter notation, if

• The element $k\in \mathbb{N}$
• The set $\mathcal{K}\supset \mathbb{N}$ is arbitrary and uncountable.

and set ${\mathsf{\Psi }}_{\omega }$ is defined as:

$${\mathsf{\Psi }}_{\omega }=\left\{\begin{array}{cc}\left\{1,···,k\right\}\hfill & \mathit{if}\mathit{there}\mathit{are}k\mathit{ways}\mathit{of}\mathit{writing}\mathit{the}\mathit{sample}\mathit{of}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \\ \mathbb{N}\hfill & \mathit{if}\mathit{there}\mathit{are}\mathit{countably}\mathit{infinite}\mathit{ways}\mathit{of}\mathit{writing}\mathit{the}\mathit{sample}\mathit{of}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \\ \mathcal{K}\hfill & \mathit{if}\mathit{there}\mathit{are}\mathit{uncountable}\mathit{ways}\mathit{of}\mathit{writing}\mathit{the}\mathit{sample}\mathit{of}\mathit{uniform}\epsilon \mathit{coverings}\mathit{of}{F}_r^{★}\hfill \end{array}\right.$$

(4.1.2)

then for every $\psi \in {\mathsf{\Psi }}_{\omega }$, the set of all samples of the set of uniform ε coverings is defined using $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$, such that ψ “enumerates" all possible samples of $\mathcal{U}(\epsilon ,F_r^{★},\omega )$.

Definition 15.

(Entropy on the sample of uniform coverings of each term of ★-sequence of sets). Since there are finitely many points in the sample of the uniform ε coverings of each term of ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ (def. 14), we:

(a)

Take a "pathway" of line segments between all points in each sample (def. 14), such that if we define the following:

i.

$\lceil ·\rceil$ is the ceiling function

ii.

$d(Q,R)$ is the Euclidean-distance between points $Q\in {\mathbb{R}}^n$ and $R\in {\mathbb{R}}^n$

iii.

The sequence:

$${\left\{x_{i-1}\right\}}_{i=1}^{\lceil H^h\left(F_r^{★}\right)/\epsilon \rceil -1}$$

contains all points in the "original" sample $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$ where we define a "pathway" for which we:

A.

Choose a point $x_0\in \mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$

B.

Take a point from $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$ (excluding $x_0$) with smallest euclidean distance from point

$x_0\in \mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$. We denote this point $x_1$ where we take $d(x_0,x_1)$. (If more than one point has the smallest Euclidean distance from $x_0$, we take either point).

C.

Take a point in $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$ (excluding $x_0$ and $x_1$) with smallest euclidean distance from $x_1$. We denote this point $x_2$, where we take $d(x_1,x_2)$. (If more than one point has the smallest Euclidean distance from $x_1$, we take either point).

D.

Take a point in $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )$ (excluding $x_0$, $x_1$, and $x_2$) with smallest euclidean distance from $x_2$. We denote this point $x_3$ then take $d(x_2,x_3)$. (If more than one point has the smallest Euclidean distance from $x_2$, we take either point).

E.

Repeat the process excluding points $x_0,x_1,x_2,x_3,$ etc. until all points in the sample are "denoted". (This should occur $\lceil H^h\left(F_r^{★}\right)/\epsilon \rceil -1$ times.)

iv.

$\mathrm{V}$ is a subset of $\left\{i\in \mathbb{N}:1\le i\le \lceil H^h\left(F_r^{*}\right)/\epsilon \rceil -1\right\}$ with the largest cardinality, where that we take the subset of i-values where $x_i$ has the $r_i$-th smallest Euclidean distance from $x_{i-1}$ (compared to every point in $\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\setminus \left\{x_{i-1}\right\}$) such that $r_i$ is not an anomaly [11] of

$$\left\{r_t:t\in \mathbb{N},1\le t\le \lceil H^h\left(F_r^{★}\right)/\epsilon \rceil -1\right\}$$

In other words:

A.

For all $w\in V$, we want $V$ to be the largest subset of $\left\{i\in \mathbb{N}:1\le i\le \lceil H^h\left(F_r^{*}\right)/\epsilon \rceil -1\right\}$ for which w-values are all i-values satisfying criteria 3(a)iv.

v.

Combining everything in $\left(3a\right)$, we ultimately want all lengths between every point in the "pathway" (def. 14) satisfying crit. 3(a)iv. We call this:

$$\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)=\left\{d(x_w,x_{w-1}):w\in \mathrm{V}\right\}$$

(b)

Using def. 15, crit. 3(a)v, normalize $\mathcal{D}$ into a discrete probability distribution. This is defined as:

$$\begin{array}{cc}\hfill & \mathbb{P}\left(\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)=\hfill \\ \hfill & \left\{y/\left(\sum _{z\in \mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)}z\right):y\in \mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(ϵ,F_r^{★},\omega ),\psi )\right)\right\}\hfill \end{array}$$

(4.1.3)

(c)

Take the entropy of (2), (for further reading, see [6], p.61-95). This is defined as:

$$E\left(\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)=-\sum _{x\in \mathbb{P}\left(\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)}x{log}_{2}x$$

(4.1.4)

(d)

Take $x_0\in \mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi ))$ where $E\left(\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)$ is maximized. Call this,

$E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)$ where:

$$\begin{array}{cc}\hfill & E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)=\hfill \\ \hfill & \underset{x_0\in \mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )}{sup}E\left(\mathcal{D}\left(x_0,{\left\{x_{w-1}\right\}}_{w\in \mathrm{V}},\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)\hfill \end{array}$$

(4.1.5)

with eq. 4.1.5 the entropy of the sample of uniform ε coverings of $F_r^{★}$.

Definition 16.

(Starred-Sequence of sets converging Sublinearly, Linearly, or Superlinearly to A compared to that of another ★-Sequence). Suppose we define starred-sequences of sets ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ and ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$, where for aconstant$\epsilon \in \mathrm{range}\left(H^h\right)$ greater than zero and variable $r\in \mathbb{N}$, we say:

(a)

Using def. 14 and 15, suppose we have:

$$\begin{array}{cc}\hfill & \underline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )|}=\hfill \\ \hfill & sup\left\{|\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })|:j\in \mathbb{N},{\omega }^{\prime }\in \Omega ,{\psi }^{\prime }\in {\Psi }_{\omega },E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })\right)\right)\le E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)\right\}\hfill \end{array}$$

(4.1.6)

then (using$\underline{|\mathcal{S}(\mathcal{U}(ϵ,F_r^{★},\omega ),\psi )|}$) we get

$$\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi ))\right|/\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underline{\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right|}$$

(4.1.7)

(b)

From def. 14 and 15, suppose we have:

$$\begin{array}{cc}\hfill & \overline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )|}=\hfill \\ \hfill & inf\left\{|\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })|:j\in \mathbb{N},{\omega }^{\prime }\in \Omega ,{\psi }^{\prime }\in {\Psi }_{\omega },E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })\right)\right)\ge E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)\right\}\hfill \end{array}$$

(4.1.8)

then (using$\overline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )|}$) we have:

$$\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right|/\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\overline{\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right|}$$

(4.1.9)

1.

If using $\overline{\alpha }\left(ϵ,r,\omega ,\psi \right)$ and $\underline{\alpha }\left(ϵ,r,\omega ,\psi \right)$ we have that:

$$\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underset{\epsilon \to 0}{lim\; sup}\underset{r\to \infty }{lim\; sup}\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=\underset{\omega \in \Omega }{inf}\underset{\psi \in {\Psi }_{\omega }}{inf}\underset{\epsilon \to 0}{lim\; inf}\underset{r\to \infty }{lim\; inf}\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=0$$

we say ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ converges to A at a rate superlinear to that of ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$.

2.

If using equations $\overline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$ and $\underline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$ (where we swap ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ in $\overline{\alpha }\left(ϵ,r,\omega ,\psi \right)$ and $\underline{\alpha }\left(ϵ,r,\omega ,\psi \right)$ with ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$) we have that:

$$\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underset{\epsilon \to 0}{lim\; sup}\underset{j\to \infty }{lim\; sup}\overline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)=\underset{\omega \in \Omega }{inf}\underset{\psi \in {\Psi }_{\omega }}{inf}\underset{\epsilon \to 0}{lim\; inf}\underset{j\to \infty }{lim\; inf}\underline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)=0$$

we then say ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ converges to A at a rate sublinear to that of ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$.

3.

If using equations $\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$, $\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$, $\overline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$, and $\underline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$ (such for the two latter, we swap

${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ in $\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ and $\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ with ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$) we haveboth:

(a)

$\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underset{\epsilon \to 0}{lim\; sup}\underset{r\to \infty }{lim\; sup}\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ or $\underset{\omega \in \Omega }{inf}\underset{\psi \in {\Psi }_{\omega }}{inf}\underset{\epsilon \to 0}{lim\; inf}\underset{r\to \infty }{lim\; inf}\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ does not equal zero

(b)

$\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underset{\epsilon \to 0}{lim\; sup}\underset{j\to \infty }{lim\; sup}\overline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$ or $\underset{\omega \in \Omega }{inf}\underset{\psi \in {\Psi }_{\omega }}{inf}\underset{\epsilon \to 0}{lim\; inf}\underset{j\to \infty }{lim\; inf}\underline{\alpha }\left(\epsilon ,j,\omega ,\psi \right)$ does not equal zeroand say ${\left(F_r^{★}\right)}_{r\in \mathbb{N}}$ converges to A at a rate linear to that of ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$.

5. Attempt to Answer Main Question Of Section 2.4

5.1. Choice Function

Suppose we define the following:

• ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ is a starred-sequence of sets (def. 8) which satisfies (1), (2), and (3) of the main question in $\S$3.4
• ${\mathbb{S}}^{\prime }\left(A\right)$ is the set of the starred-sequences of sets that have finite generalized mean (def. 9).
• ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$ is an element ${\mathbb{S}}^{\prime }\left(A\right)$ butnotan element in the set of equivalent starred-sequences of sets (def. 11) of ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ where using note 3.4, we can represent this criteria as:

  $${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}\in {\mathbb{S}}^{\prime }\left(A\right)\setminus \sim {\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$$

  (5.1.1)

Further note, from def. 16, if we take:

$$\begin{array}{cc}\hfill & \overline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )|}=\hfill \\ \hfill & inf\left\{|\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })|:j\in \mathbb{N},{\omega }^{\prime }\in \Omega ,{\psi }^{\prime }\in {\Psi }_{\omega },E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })\right)\right)\ge E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )\right)\right)\right\}\hfill \end{array}$$

(5.1.2)

and from def. 16, we take:

$$\begin{array}{cc}\hfill & \underline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )|}=\hfill \\ \hfill & sup\left\{|\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })|:j\in \mathbb{N},{\omega }^{\prime }\in \Omega ,{\psi }^{\prime }\in {\Psi }_{\omega },E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},{\omega }^{\prime }),{\psi }^{\prime })\right)\right)\le E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )\right)\right)\right\}\hfill \end{array}$$

(5.1.3)

Then, when we write def. 14, eq. 5.1.2 and eq. 5.1.3 as:

$$\begin{array}{c}\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}|\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )|=|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})|=|{\mathcal{S}}^{\prime }|\end{array}$$

(5.1.4)

$$\begin{array}{c}\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\overline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )|}=\overline{|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})|}=\overline{|{\mathcal{S}}^{\prime }|}\end{array}$$

(5.1.5)

$$\begin{array}{c}\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underline{|\mathcal{S}(\mathcal{U}(\epsilon ,F_k^{★★★},\omega ),\psi )|}=\underline{|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})|}=\underline{|{\mathcal{S}}^{\prime }|}\end{array}$$

(5.1.6)

the choice function (which we’ll later define on pg. 24, thm. 4) should immediately choose $F_k^{★★★}$ when

• For all $m\in \left\{1,···,n-1\right\}$ when defining the set of all values of the m-th coordinate of $(c_1,c_2,···,c_n)\in F_k^{★★★}$ (i.e., $F_{k,m}^{★★★}$—where, unlike cit.[1, §4], we focus on the domain of $F_k^{★★★}$ to get "$n-1$" instead of "n"), then when $z>0$, we either want:

  (a)

  $sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)=z$ and $inf\left(F_{k+1,m}^{★★★}\right)-inf\left(F_{k,m}^{★★★}\right)=-z$.

  (b)

  $sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)=0$ and $inf\left(F_{k+1,m}^{★★★}\right)-inf\left(F_{k,m}^{★★★}\right)=-z$.

  (c)

  $sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)=z$ and $inf\left(F_{k+1,m}^{★★★}\right)-inf\left(F_{k,m}^{★★★}\right)=0$.

  (d)

  $sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)=0$ and $inf\left(F_{k+1,m}^{★★★}\right)-inf\left(F_{k,m}^{★★★}\right)=0$.

• If the center of the universe is a chosen point $Z\in {\mathbb{R}}^n$, where:

  $$Z=(z_1,z_2,···,z_n)$$

  (5.1.7)
  then for all $m\in \left\{1,···,n-1\right\}$, there exists $q\in \mathbb{N}$, s.t.for all $k\ge q$, when set $F_{k,m}^{★★★}$ is a collection of all the values of the m-th co-ordinate of $(c_1,c_2,···,c_n)\in F_k^{★★★}$ (again, unlike cit.[1, §4], we focus on the domain of $F_k^{★★★}$ to get "$n-1$" instead of "n"), we must get:

  $$\frac{1}{H^h\left(F_{k,m}^{★★★}\right)}{\int }_{F_{k,m}^{★★★}}xd{H}^h=z_m$$

  (5.1.8)

where, using absolute value function $\left||·|\right|$ and $m\in \left\{1,2,···,n\right\}$, when set $F_{k,m}^{★★★}$ is a collection of all the values of the m-th co-ordinate of $(c_1,c_2,···,c_n)\in F_k^{★★★}$, for $z>0$, when we define:

$$\begin{array}{cccc}\hfill & S(z,k,m)=\left|\right|z\hfill & & -\left(sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)\right)\left(inf\left(F_{k,m}^{★★★}\right)-inf\left(F_{k+1,m}^{★★★}\right)\right)\hfill \\ & & & \left|\right|\left(inf\left(F_{k,m}^{★★★}\right)-inf\left(F_{k+1,m}^{★★★}\right)\right)\left(sup\left(F_{k+1,m}^{★★★}\right)-sup\left(F_{k,m}^{★★★}\right)-1\right)\left|\right|\left|\right|\hfill \end{array}$$

(5.1.9)

and

$$\begin{array}{cccc}\hfill & T(z_m,k,m)=\hfill & & \left[\left(sup\left(F_{k+1,m}^{★★★}\right)-z_m\right)\left(inf\left(F_{k,m}^{★★★}\right)-z_m\right)-\left(sup\left(F_{k,m}^{★★★}\right)-z_m\right)\left(inf\left(F_{k+1,m}^{★★★}\right)-z_m\right)\right]\hfill \\ & & & \left[\left(inf\left(F_{k,m}^{★★★}\right)-z_m\right)-\left(inf\left(F_{k+1,m}^{★★★}\right)-z_m\right)+\left(sup\left(F_{k+1,m}^{★★★}\right)-z_m\right)-\left(sup\left(F_{k,m}^{★★★}\right)-z_m\right)-1\right]\hfill \\ & & & \left[\left(inf\left(F_{k,m}^{★★★}\right)-z_m\right)-\left(inf\left(F_{k+1,m}^{★★★}\right)-z_m\right)\right]\left[\left(sup\left(F_{k+1,m}^{★★★}\right)-z_m\right)-\left(sup\left(F_{k,m}^{★★★}\right)-z_m\right)\right]\hfill \end{array}$$

(5.1.10)

criteria (1) is achieved, using eq. 5.1.9, when:

$$S^{\prime }(z,k)=\frac{1}{n-1}\sum _{m=1}^{n-1}S(z,k,m)$$

(5.1.11)

such that, for all $k\in \mathbb{N}$:

$$S^{\prime }(z,k)=1$$

(5.1.12)

and criteria (2) is achieved, using eq. 5.1.7 and 5.1.10, when:

$$T^{\prime }(Z,k)=\frac{1}{n-1}\sum _{m=1}^{n-1}T(z_m,k,m)$$

(5.1.13)

such that, for all $k\in \mathbb{N}$:

$$T^{\prime }(Z,k)=0$$

(5.1.14)

where we consider the following:

5.2. Question:

How do we create a choice function which solves the question in sec. 3.4 using ${\mathcal{S}}^{\prime }$, $\overline{|{\mathcal{S}}^{\prime }|}$, $\underline{|{\mathcal{S}}^{\prime }|}$, $S^{\prime }(z,k)$, and $T^{\prime }(Z,k)$ or equations 5.1.4, 5.1.5, 5.1.6, 5.1.11 and 5.1.13 resp.?

5.3. "Attempt" to answer the Question

(Note the attempt might be wrong but could offer hints to how the solution would appear).

Suppose $z=1$ and the chosen coordinate for the center of the universe (i.e., eq. 5.1.7) is the origin, where $z_m=0$ for all $m\in \left\{1,···,n\right\}$:

$$\begin{array}{cc}\hfill & Z=(z_1,z_2,···,z_n)\Rightarrow \hfill \\ \hfill & O=\left(\underset{n\mathit{times}}{\underbrace{0,0,···,0}}\right)\hfill \end{array}$$

(5.3.1)

Using equations ${\mathcal{S}}^{\prime }$, $\overline{|{\mathcal{S}}^{\prime }|}$, $\underline{|{\mathcal{S}}^{\prime }|}$, $S^{\prime }(z,k)$, and $T^{\prime }(Z,k)$ (i.e., eq. 5.1.4, 5.1.5, 5.1.6, 5.1.11 and 5.1.13) with the nearest integer function $\left[·\right]$, we define:

$$\begin{array}{cc}& K(\epsilon ,F_k^{★★★})=\hfill \\ & S^{\prime }(1,k)\left(\left|\right|\frac{|{\mathcal{S}}^{\prime }|\left(1+\left[\frac{|{\mathcal{S}}^{\prime }|\left(\underline{|{\mathcal{S}}^{\prime }|}+2\left|{\mathcal{S}}^{\prime }\right|\right)}{\left(\underline{|{\mathcal{S}}^{\prime }|}+\left|{\mathcal{S}}^{\prime }\right|\right)\left(\underline{|{\mathcal{S}}^{\prime }|}+\left|{\mathcal{S}}^{\prime }\right|+\overline{|{\mathcal{S}}^{\prime }|}\right)}\right]\right)\left(1+\left[\underline{|{\mathcal{S}}^{\prime }|}/\left|{\mathcal{S}}^{\prime }\right|\right]\right)}{\left(1+\left[|{\mathcal{S}}^{\prime }|/\overline{|{\mathcal{S}}^{\prime }|}\right]\right)\left(1+\left[\underline{|{\mathcal{S}}^{\prime }|}/\overline{|{\mathcal{S}}^{\prime }|}\right]\right)}-|{\mathcal{S}}^{\prime }\left|\right||+|{\mathcal{S}}^{\prime }|\right)-T^{\prime }(O,k)\hfill \end{array}$$

(5.3.2)

where using $K(\epsilon ,F_k^{★★★})$, the choice function should be the following:

Theorem 4.

If we define:

$$\mathcal{M}(\epsilon ,F_k^{★★★})=\left|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})\right|\left(K(\epsilon ,F_k^{★★★})-\left|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})\right|\right)$$

$$\mathcal{M}(\epsilon ,F_j^{★★})=\left|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})\right|\left(K(\epsilon ,F_j^{★★})-\left|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})\right|\right)$$

where for $\mathcal{M}(\epsilon ,F_k^{★★★})$, we define $\mathcal{M}(\epsilon ,F_k^{★★★})$ to be the same as $\mathcal{M}(\epsilon ,F_j^{★★})$ when swapping "$j\in \mathbb{N}$" with "$k\in \mathbb{N}$" (for eq. 5.1.5 & 5.1.6) and sets $F_k^{★★★}$ with $F_j^{★★}$ (for eq. 5.1.4–5.3.2), then for constant $v>0$ and variable $v^{*}>0$, if:

$$\overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})=inf\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\ge \mathcal{M}(\epsilon ,F_k^{★★★})\ge v^{*}\right\}\cup \left\{v^{*}\right\}\right)+v$$

(5.3.3)

and:

$$\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})=sup\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},v^{*}\le \mathcal{M}(\epsilon ,F_j^{★★})\le \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\cup \{-v^{*}\}\right)+v$$

(5.3.4)

then for all ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}\in {\mathbb{S}}^{\prime }\left(A\right)\setminus \sim {\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ ($\S$, crit. 3), if:

$$\begin{array}{cc}\hfill & \underset{\epsilon \to 0}{lim\; inf}\underset{v^{*}\to \infty }{lim}\underset{k\to \infty }{lim}\frac{|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})|+v}{\overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})}=\hfill \\ \hfill & \underset{\epsilon \to 0}{lim\; sup}\underset{v^{*}\to \infty }{lim}\underset{k\to \infty }{lim}\frac{|{\mathcal{S}}^{\prime }(\epsilon ,F_k^{★★★})|+v}{\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})}=0\hfill \end{array}$$

(5.3.5)

we choose ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ satisfying eq. 5.3.5. (Note, we want $sup\varnothing =-\infty$, $inf\varnothing =+\infty$, and ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ to answer the main question of $\S$3.4) where the answer to the focus3 is ${\mathbb{E}}^{**}[f,F_k^{★★★}]$ in eq. ?? (when it exists):

$$\begin{array}{cc}\hfill & \forall (ϵ>0)\exists (N\in \mathbb{N})\forall (k\in \mathbb{N})\left(k\ge N\Rightarrow \left|\frac{1}{H^h\left(\mathrm{dom}\left(F_k^{★★★}\right)\right)}{\int }_{\mathrm{dom}\left(F_k^{★★★}\right)}{f}_k^{★}d{H}^h-{\mathbb{E}}^{**}[f,F_k^{★★★}]\right|<ϵ\right)\hfill \end{array}$$

(5.3.6)

[Explanation of Theorem 4]

The theorem 4 is similar to the methods used in def. 16 crit. 0a and 0b or $\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ and $\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)$ and def. 16 crit. 1, where:

$$\underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underset{\epsilon \to 0}{lim\; sup}\underset{r\to \infty }{lim\; sup}\overline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=\underset{\omega \in \Omega }{inf}\underset{\psi \in {\Psi }_{\omega }}{inf}\underset{\epsilon \to 0}{lim\; inf}\underset{r\to \infty }{lim\; inf}\underline{\alpha }\left(\epsilon ,r,\omega ,\psi \right)=0$$

such that we replace:

$$\begin{array}{cc}\hfill & E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right)\right)\mapsto \mathcal{M}(\epsilon ,F_k^{★★★})\hfill \\ \hfill & E\left(\mathcal{D}\left(\mathcal{S}(\mathcal{U}(\epsilon ,F_j^{★★},\omega ),\psi )\right)\right)\mapsto \mathcal{M}(\epsilon ,F_j^{★★})\hfill \\ \hfill & \left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi ))\right|\mapsto \left|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})\right|\hfill \\ \hfill & \underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\underline{\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right|}\mapsto \underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})\hfill \\ \hfill & \underset{\omega \in \Omega }{sup}\underset{\psi \in {\Psi }_{\omega }}{sup}\overline{\left|\mathcal{S}(\mathcal{U}(\epsilon ,F_r^{★},\omega ),\psi )\right|}\mapsto \overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})\hfill \end{array}$$

note the changes to def. 16, crit. 2 were made, so $\mathcal{M}(\epsilon ,F_k^{★★★})$ is "large enough" compared to $\mathcal{M}(\epsilon ,F_j^{★★})$, with ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}$ non-equivalent to ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ (e.g. when $A=\mathbb{Q}$, ${\left(F_k^{★★★}\right)}_{k\in \mathbb{N}}$ should be ${\left(\left\{c/k!:c\in \mathbb{N},1\le c\le k!\right\}\right)}_{k\in \mathbb{N}}$ and never give $\mathcal{M}(\epsilon ,F_k^{★★★})$ smaller than "small" $\mathcal{M}(\epsilon ,F_j^{★★})$, e.g.:

$${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}={\left(\left\{u/w:u\in \mathbb{Z},w\in \mathbb{N},w\le j,-w·j\le u\le w·j\right\}\right)}_{j\in \mathbb{N}}$$

or larger than "large" $\mathcal{M}(\epsilon ,F_j^{★★})$; e.g., ${\left(F_j^{★★}\right)}_{j\in \mathbb{N}}={\left(\left\{u_1/\left(6(j!)\right):u_1\in \mathbb{Z},-6j·j!\le u_1\le 6j·j!\right\}\right)}_{j\in \mathbb{N}}$)

Moreover, in $\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ and $\overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ of thm. 4, we add constant $v>0$ and variable $v^{*}>0$ so if either

• $\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})-v=0$ (i.e., using a related limit to eq. 5.3.5, division by zero is undefined).
• $\overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})-v=0$ (i.e., using a related limit to eq. 5.3.5, division by zero is undefined).
• $inf\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\ge \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)=+\infty$ (i.e., similar to $\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ of eq. 5.3.3, with no variable $v^{*}$ such that $\mathcal{M}(\epsilon ,F_k^{★★★})=0$ and $\exists (J>0)\forall (j_1>0)\exists (j\ge j_1)(\mathcal{M}(\epsilon ,F_j^{★★})\le J)$, where we apply a related limit to eq. 5.3.5 that’s undefined due to division by infinity.)
• $inf\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\ge \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)=\varnothing$ (i.e., similar to $\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ of eq. 5.3.3, with no variable $v^{*}$ and $\mathcal{M}(\epsilon ,F_j^{★★})=0$, where we apply a related limit to eq. 5.3.5 that’s undefined since
  $inf\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\ge \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)$ is an undefined empty set.)
• $sup\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\le \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)=+\infty$ (i.e., similar to $\overline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ of eq. 5.3.4, with no variable $v^{*}$ and $\mathcal{M}(\epsilon ,F_j^{★★})=0$, where we apply a related limit to eq. 5.3.5 that’s undefined due to division by infinity.)
• $sup\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\le \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)=\varnothing$ (i.e., similar to $\underline{\mathcal{S}}(\epsilon ,k,v^{*},F_j^{★★})$ of eq. 5.3.3, with no variable $v^{*}$ and $\mathcal{M}(\epsilon ,F_k^{★★})=0$, where we apply a related limit to eq. 5.3.5 that’s undefined since$inf\left(\left\{|{\mathcal{S}}^{\prime }(\epsilon ,F_j^{★★})|:j\in \mathbb{N},\mathcal{M}(\epsilon ,F_j^{★★})\ge \mathcal{M}(\epsilon ,F_k^{★★★})\right\}\right)$ is an undefined empty set.)
• $\left|\right\{z:j,z\in \mathbb{N},\mathcal{M}(\epsilon ,F_{j+z})\le \mathcal{M}(\epsilon ,F_j)\left\}\right|=+\infty$ (i.e., infinite number succeeding $F_j$ are smaller than original $F_j$, where such $F_j$ should be eliminated).

the limit in eq. 5.3.5 still exists.

5.4. Question:

How do we use mathematica code to illustrate §4 and 5?