Concept Paper
Version 20
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Meaningfully Averaging Unbounded Functions
Version 1
: Received: 9 July 2023 / Approved: 10 July 2023 / Online: 10 July 2023 (08:56:52 CEST)
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Version 12 : Received: 22 December 2023 / Approved: 25 December 2023 / Online: 27 December 2023 (09:33:43 CET)
Version 13 : Received: 28 December 2023 / Approved: 28 December 2023 / Online: 29 December 2023 (03:06:03 CET)
Version 14 : Received: 30 December 2023 / Approved: 3 January 2024 / Online: 3 January 2024 (05:33:34 CET)
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Version 20 : Received: 26 March 2024 / Approved: 26 March 2024 / Online: 27 March 2024 (09:10:36 CET)
How to cite: Krishnan, B. Meaningfully Averaging Unbounded Functions. Preprints 2023, 2023070560. https://doi.org/10.20944/preprints202307.0560.v20 Krishnan, B. Meaningfully Averaging Unbounded Functions. Preprints 2023, 2023070560. https://doi.org/10.20944/preprints202307.0560.v20
Abstract
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.
Keywords
expected value; hausdorff measure; (Exact) dimension function; function space; prevalent and shy sets; entropy; choice function
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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