Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Bharath Krishnan

doi:10.20944/preprints202307.0560.v17

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preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202307.0560.v17

Preprint Concept Paper Version 17 Preserved in Portico This version is not peer-reviewed

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Bharath Krishnan

Version 1 : Received: 9 July 2023 / Approved: 10 July 2023 / Online: 10 July 2023 (08:56:52 CEST)
Version 2 : Received: 10 July 2023 / Approved: 11 July 2023 / Online: 11 July 2023 (09:34:34 CEST)
Version 3 : Received: 11 July 2023 / Approved: 12 July 2023 / Online: 13 July 2023 (05:00:22 CEST)
Version 4 : Received: 13 July 2023 / Approved: 13 July 2023 / Online: 14 July 2023 (05:13:22 CEST)
Version 5 : Received: 17 July 2023 / Approved: 17 July 2023 / Online: 17 July 2023 (10:02:07 CEST)
Version 6 : Received: 17 July 2023 / Approved: 18 July 2023 / Online: 19 July 2023 (03:39:53 CEST)
Version 7 : Received: 21 July 2023 / Approved: 21 July 2023 / Online: 21 July 2023 (09:36:51 CEST)
Version 8 : Received: 22 July 2023 / Approved: 24 July 2023 / Online: 24 July 2023 (10:51:59 CEST)
Version 9 : Received: 12 December 2023 / Approved: 13 December 2023 / Online: 13 December 2023 (10:12:55 CET)
Version 10 : Received: 13 December 2023 / Approved: 15 December 2023 / Online: 15 December 2023 (04:57:26 CET)
Version 11 : Received: 19 December 2023 / Approved: 21 December 2023 / Online: 22 December 2023 (08:47:53 CET)
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)
Version 15 : Received: 3 January 2024 / Approved: 4 January 2024 / Online: 4 January 2024 (09:37:33 CET)
Version 16 : Received: 4 January 2024 / Approved: 5 January 2024 / Online: 5 January 2024 (10:07:41 CET)
Version 17 : Received: 9 February 2024 / Approved: 10 February 2024 / Online: 12 February 2024 (12:09:52 CET)
Version 18 : Received: 18 March 2024 / Approved: 19 March 2024 / Online: 19 March 2024 (12:46:40 CET)
Version 19 : Received: 23 March 2024 / Approved: 25 March 2024 / Online: 26 March 2024 (08:22:18 CET)
Version 20 : Received: 26 March 2024 / Approved: 26 March 2024 / Online: 27 March 2024 (09:10:36 CET)

How to cite: Krishnan, B. Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions. Preprints 2023, 2023070560. https://doi.org/10.20944/preprints202307.0560.v17 Krishnan, B. Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions. Preprints 2023, 2023070560. https://doi.org/10.20944/preprints202307.0560.v17

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MDPI and ACS Style

Krishnan, B. Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions. Preprints 2023, 2023070560. https://doi.org/10.20944/preprints202307.0560.v17

APA Style

Krishnan, B. (2024). Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions. Preprints. https://doi.org/10.20944/preprints202307.0560.v17

Chicago/Turabian Style

Krishnan, B. 2024 "Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions" Preprints. https://doi.org/10.20944/preprints202307.0560.v17

Abstract

Suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; we want to find an unique and “natural” extension of the expected value, w.r.t the Hausdorff measure on bounded functions to unbounded/bounded f, that's finite for all f in a non-shy subset of B*---the set of all Borel measurable functions in ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of B*. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, 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, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique and “natural” extension for all f in a non-shy subset of B*. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.

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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