Preprint Concept Paper Version 15 Preserved in Portico This version is not peer-reviewed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
* ORCID logo
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.v15 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.v15

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, 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 (defined on §2.3 & §2.4) 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.

Download PDF

Comments (1)

Comment 1
Received: 4 January 2024
Commenter: Bharath Krishnan
Commenter's Conflict of Interests: Author
Comment: Minor edits---Fixed formatting between def. 11 & fig. 1. 

Made changes to def. 19.

In abstract, changed ℝᴬ to B*.
+ Respond to this comment

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 1
Get PDF Cite Share 0
Bookmark BibSonomy Mendeley Reddit Delicious
×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.