preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202307.0560.v20

Preprint Concept Paper Version 20 Preserved in Portico This version is not peer-reviewed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". For any n∈ ℕ, set A⊆ℝⁿ and set B⊆ℝ where (A,P) is a Polish space, we illustrate this quote with an everywhere, surjective function f∶A→B. The problem is no meaningful expected value of f (e.g., w.r.t the Lebesgue or Hausodorff measure) on Borel sets has a finite value, since the graph of f in any n-dim. interval which covers a subset of A×B has countably infinite points. (The Hausdorff measure of countably infinite points is +∞, where the expected value of f is undefined due to division by infinity.) To fix this, we need the most generalized and "meaningful" expected value; however, consider the following issue. 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-valued expected value—w.r.t the Hausdorff measure—then B** is a shy subset of B*. Hence a "positive measure" of Borel measurable functions needs to have a finite expected value to increase the chance that everywhere, surjective f:A→B has a finite expectation. To fix this 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. Note, we haven’t found evidence suggesting mathematicians 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 choose a sequence of bounded functions using a "choice function". Note, we find the "choice function" using a question with criteria in §2.4. Also, in §3 and §4, we attempt to answer this question that should "choose" a sequence of bounded functions that a) meaningfully averages everywhere, surjective functions and b) obtains a finite average from a "positive measure" of Borel measurable functions.

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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