preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202307.0560.v19

Preprint Concept Paper Version 19 Preserved in Portico This version is not peer-reviewed

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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 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x₁,...,xₙ,fₖ*(x₁,...,xₙ))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, if (Fₖ***) converges to A at a rate linear or super-linear to the rate non-equivelant sequences of sets converge, and the extended expected value of or E**[f,Fₖ***] in the eq. below: ∀(ε>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

expected value; hausdorff measure; (Exact) dimension function; function space; prevalent and shy sets; entropy; choice function

Computer Science and Mathematics, Analysis

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