Motivation: Suppose, we partition $\mathbb{R}$ into sets $A$ and $B$ with a positive measure in each non-empty, open interval. I want a simple example of a piece-wise function $f:\mathbb{R} \to \mathbb{R}$, where $\mathscr{G_1}:A \to \mathbb{R}$ and $\mathscr{G}_2:B \to\mathbb{R}$, such that: $$f(x)=\begin{cases} \mathscr{G}_1(x) & x\in A\\ \mathscr{G}_2(x) & x\notin A \Rightarrow x\in\mathbb{R}\setminus A \Rightarrow x\in B \end{cases}$$ where $f$ is simple to average with this paper:
We take chosen sequences of bounded functions converging to $f$ with the same satisfying and finite expected value w.r.t. a reference point, the rate of expansion of a sequence of each bounded function’s graph, and a “measure” of each bounded function's graph involving covers, samples, pathways, and entropy.
However, we want some function $f$, where using $\S$2.3.1 of this paper:
$\S$2.3.1 (Sequences of Functions Converging to $f$)
A sequence of functions $(f_r)_{r\in\mathbb{N}}$, where $(A_r)_{r\in\mathbb{N}}$ is a sequence of sets and functions $f_{r}:A_r\to\mathbb{R}$, converges to a function $f:\mathbb{R}\to\mathbb{R}$ when:
For any $x\in \mathbb{R}$, there exists a sequence $x_r\in A_r$ s.t. $x_r\to x$ and $f_r(x_r)\to f(x)$.
This is equivelant to:
$(f_r,A_r)\to (f,\mathbb{R})$
$(f_r,A_r)\to (f,\mathbb{R})$ and $(g_v,B_v)\to(f,\mathbb{R})$ but using $\S$2.3.2 of this paper:
$\S$2.3.2 (Expected Value of Sequences of Functions Converging to $f$)
Suppose, we define:
- $(f_r,A_r)\to(f,\mathbb{R})$
- $|\cdot|$ 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 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.
we get $\mathbb{E}[f_r]\neq \mathbb{E}[g_v]$.
Question: In $\S$3.1 of this paper, suppose $C=(0,0)$ and $E=1/2$. How do we define $f$, where $\mathbb{E}[f_r^{\star}]\neq 0$?
Attempt: The only example I know of is the following:
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\dim_{\text{H}}(\cdot)$ be the Hausdorff dimension, where $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.
If $G$ is the graph of $f$, we want an explicit $f$, such that:
- The function $f$ is everywhere surjective (i.e., $f[(a,b)]=\mathbb{R}$ for all non-empty open intervals $(a,b)$)
- $\mathcal{H}^{\dim_{\text{H}}(G)}(G)=0$
An explicit example of $f$ is defined here; however, finding $\mathbb{E}[f_r^{\star}]$ using $\S$3.1 of this paper is difficult and $f$ is not the same as the $f$ defined in the motivation.
