Meaningfully Averaging Unbounded Functions
Bharath S. Krishnan · Preprints.org · 2024
In this paper we want meanignfully average an "infinite collection of objects covering an infinite expanse of space". We illustrate this quote with an explicit n-dimensional function, where the graph of the function is dense in ℝ^(n+1). The problem is non meaningful expected value of the function (e.g., w.r.t the Lebesgue or Hausodorff measure) has a finite value. In fact "almost no" Borel measurable functions have a finite and meaningful value. In rigorous terms, suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; B* is the set of all Borel measurable function in ℝᴬ for all A⊆ℝⁿ, and B** is the set of all f∈B* with a finite expected value—w.r.t the Hausdorff measure—then B** is a shy subset of B*. To fix these issues, we wish to find a unique and "natural" extension of the expected value—w.r.t the Hausdorff measure—on bounded functions to unbounded/bounded f, which takes finite values only, so B** is a non-shy subset of B*. Note we haven't found evidence suggesting mathematicians have thought of this problem; however, it's assumed, in general, there's no meaningful way of averaging functions which cover an infinite expanse of space. Regardless, we'll attempt to solve the problem by defining a choice function—this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) exists, then E**[f,Fₖ***] is a unique and "natural" extension of the original expected value on bounded f, so B**—the set of all f∈B* with a finite E**[f,Fₖ***]—is a non-shy set in B*. Note we guessed the choice function using computer programming we redefine linear and super-linear convergence in terms of partitions of