What is an applicable way of averaging an everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure?
Motivation:
In a magazine article on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known math 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.”
According to this answer, the path integral averages a set of functions matching Wood’s description rather than a real function whose graph also matches Wood’s description. Regardless, solving the latter can have applications in physics. (Note, if we can average the function below, we can average any real function.)
$\S$1. Example To Average:
Suppose $f:\mathbb{R}\to\mathbb{R}$ is an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure. (The function aligns with the motivation.)
Consider a Cantor set $\mathcal{C}\subseteq[0,1]$ with Hausdorff dimension $0$. Now consider a countable disjoint union $\cup_{m\in\mathbb{N}}\,\mathcal{C}_m$ such that each $\mathcal{C}_m$ is the image of $\mathcal{C}$ by some affine map and every open set $O\subseteq[0,1]$ contains $\mathcal{C}_m$ for some $m$. Such a countable collection can be obtained by e.g. at letting $\mathcal{C}_{m}$ be contained in the biggest connected component of $[0,1]\setminus(\mathcal{C}_1\cup\dots\cup \mathcal{C}_{m-1})$ (with the center of $\mathcal{C}_m$ being the middle point of the component).
Note that $\cup_m \mathcal{C}_m$ has Hausdorff dimension $0$, so $\left(\cup_m \mathcal{C}_m\right)\times[0,1]\subseteq\mathbb{R}^2$ has Hausdorff dimension one.
Now, let $g:[0,1]\to\mathbb{R}$ such that $g|_{\mathcal{C}_m}$ is a bijection $\mathcal{C}_m\to\mathbb{R}$ for all $m$ (all of them can be constructed from a single bijection $\mathcal{C}\to\mathbb{R}$, which can be obtained without choice, although it may be ugly to define) and outside $\cup_m \mathcal{C}_m$ let $g$ be defined by $g(x)=h(x)$, where $h:[0,1]\to\mathbb{R}$ has a graph with Hausdorff dimension $2$ (this doesn't require choice either).
Then, the function $g$ has a graph with Hausdorff dimension $2$ and is everywhere surjective, but its graph has Lebesgue measure $0$ because it is a graph (so it admits uncountably many disjoint vertical translates).
Note, we can make the construction with union of $\mathcal{C}_m$ rather explicit as follows. Split the binary expansion of $x$ as strings of size with a power of two, say $x=0.1101000010\ldots$ becomes $(s_0,s_1,s_2,\ldots)=(1,10,1000,\ldots)$. If this sequence eventually contains only strings of the form $0\cdots0$ or $1\cdots 1$, say after $s_k$, then send it to $y=\sum_{i>0}\epsilon_{i}2^{-i}$, where $s_{k+i}=\epsilon_i\cdots\epsilon_i$. Otherwise, send it to the explicit continuous function $h$ given by the linked article. This will give you something from $[0,1)\to[0,1)$
Finally, compose an explicit (reasonable) bijection from $[0,1)$ to $\mathbb{R}$. In this case, the construction can be easily adapted so that the $[0,1]$ or $[0,1)$ target space is actually $(0,1)$, then compose with $t\mapsto(1-2x)/(x^2-x)$.
While reading this post, consider the following:
Question: Similar to my attempt in Section $\S$3, how does one define a leading question that chooses a "satisfying" and finite average for an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure?
$\S$2. Problem With "Current Mean" And Its Replacement
Since the integral of $f$ w.r.t. Hausdorff measure in its dimension is undefined (i.e., the graph of $f$ has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure), the expected value of $f$ is undefined.
Thus, take the mean of a sequence of bounded functions with different domains converging to $f$ (when it exists). For the sake of application, we want the mean to be finite.
The problem is depending on the sequence of bounded functions chosen, the expected value of a sequence of bounded functions can be one of several values (when it exists). In fact, the set of all $f\in\mathbb{R}^{\mathbb{R}}$—where two sequences of bounded functions converging to $f$ have non-equivalent expected values (when either exist)—forms a prevalent “full measure” subset of $\mathbb{R}^{\mathbb{R}}$.
Definition 2.1 (Prevalent/Shy Sets)
Let $X$ be a completely metrizable topological space. A Borel set $E\subset X$ is said to be prevalent if there exists a Borel 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.
Moreover:
- The complement of a prevalent set is a shy set
Hence:
- If $F\subset X$ is prevalent, 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 $F$.
In the example of Section $\S$1, the expected value of a sequence of bounded functions converging to $f$ is one of several values (when it exists): that is, since the center of a sequence of each bounded function's graph would vary—depending on the sequence of bounded functions chosen—the expected value of this sequence is any value w.r.t. the 2nd coordinate of the center. (We call the "center" the reference point $\mathbf{R}\in\mathbb{R}^2$.)
$\S$3. Solution to the Problems in $\S$2 Using an Answer to a Question
Therefore, we need a useful way of choosing a “satisfying” expected value of $f\in\mathbb{R}^{\mathbb{R}}$ that is finite for a prevelant (Definition 2.1) subset of $\mathbb{R}^{\mathbb{R}}$. There are many ways but one involves an answer to a leading question (i.e., using a choice function). Note that this should have an application in physics.
For instance, the leading question can be defined w.r.t. four criteria:
- the chosen sequences of bounded functions, which converge to an arbitrary $\mathsf{f}\in\mathbb{R}^{\mathbb{R}}$, have the same finite expected value. (This means that the chosen sequences are equivalent to each other. However, when there exists a $\mathsf{f}\in\mathbb{R}^{\mathbb{R}}$ where a sequence of bounded functions converging to $\mathsf{f}$ have a non-equivalent expected value compared to the expected value of each chosen sequence, the sequence then is non-equivalent to the chosen sequences. Moreover, if one chosen sequence out of all chosen sequences satisfy a criteria, then so do all the other chosen sequences.)
- If 1. is true, the metric entropy (see Edit 1) of the chosen sequence of each bounded function's graph increases at a rate linear or superlinear to compared to that of every "non-equivalent" sequence of each bounded function's graph (i.e., the chosen "non-equivalent" sequences of bounded functions satisfy 1.).
- Edit 1: If criteria 2. is unclear, see the original paper (i.e., "families" are a generalization of sequences and the metric entropy is assumed to be the "measure"). If the assumptions are incorrect, please correct me.
- If 1. and 2. are true, the absolute difference between the 2nd coordinate of the reference point $\mathbf{R}\in\mathbb{R}^{2}$ and the expected value of the chosen sequence of bounded functions converging to $f$ is minimized w.r.t. the same measurement of every chosen sequence of bounded functions satisfying 1. and 2.
- If 1., 2., and 3. are true, the absolute difference between the expected rate of expansion and the actual rate of expansion of the chosen sequence of each bounded function's graph is minimized w.r.t. the same measure of every chosen sequence of each bounded function's graph (i.e., the chosen "non-equivalent" sequences of bounded functions satisfy 1., 2., and 3.).
Restated Question: Similar to my attempt, how does one define a leading question that chooses a "satisfying" and finite average for an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure. (Use the four criteria.)
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