Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions

Bharath Krishnan

doi:10.20944/preprints202302.0367.v7

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preprints.org > computer science and mathematics > mathematics > doi: 10.20944/preprints202302.0367.v7

Preprint Concept Paper Version 7 Preserved in Portico This version is not peer-reviewed

Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions

Bharath Krishnan

Version 1 : Received: 21 February 2023 / Approved: 22 February 2023 / Online: 22 February 2023 (02:06:52 CET)
Version 2 : Received: 23 February 2023 / Approved: 23 February 2023 / Online: 23 February 2023 (03:45:21 CET)
Version 3 : Received: 28 February 2023 / Approved: 28 February 2023 / Online: 28 February 2023 (04:56:00 CET)
Version 4 : Received: 1 March 2023 / Approved: 1 March 2023 / Online: 1 March 2023 (07:17:23 CET)
Version 5 : Received: 9 March 2023 / Approved: 9 March 2023 / Online: 9 March 2023 (06:42:54 CET)
Version 6 : Received: 11 March 2023 / Approved: 14 March 2023 / Online: 14 March 2023 (01:38:37 CET)
Version 7 : Received: 17 March 2023 / Approved: 17 March 2023 / Online: 17 March 2023 (04:04:07 CET)
Version 8 : Received: 30 March 2023 / Approved: 30 March 2023 / Online: 30 March 2023 (02:46:43 CEST)
Version 9 : Received: 4 April 2023 / Approved: 6 April 2023 / Online: 6 April 2023 (10:04:39 CEST)
Version 10 : Received: 6 April 2023 / Approved: 7 April 2023 / Online: 7 April 2023 (05:16:10 CEST)
Version 11 : Received: 25 April 2023 / Approved: 26 April 2023 / Online: 26 April 2023 (03:35:26 CEST)

How to cite: Krishnan, B. Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions. Preprints 2023, 2023020367. https://doi.org/10.20944/preprints202302.0367.v7 Krishnan, B. Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions. Preprints 2023, 2023020367. https://doi.org/10.20944/preprints202302.0367.v7

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MDPI and ACS Style

Krishnan, B. Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions. Preprints 2023, 2023020367. https://doi.org/10.20944/preprints202302.0367.v7

APA Style

Krishnan, B. (2023). Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions. Preprints. https://doi.org/10.20944/preprints202302.0367.v7

Chicago/Turabian Style

Krishnan, B. 2023 "Defining the Most Generalized, Natural Extension of the Expected Value on Measurable Functions" Preprints. https://doi.org/10.20944/preprints202302.0367.v7

Abstract

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on sets measurable in the Caratheodory sense to be finite for a larger class of functions, since the set of all measurable functions with infinite or undefined expected values may form a prevalent subset of the set of all measurable functions. This means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, with a unique solution that allows "more" functions to have finite expected values, we'll outline some preliminary definitions. We'll then define the specific problem in section 2 (with a partial solution in section 3) to visualize the complete solution to the problem. Along the way, we will ask a series of questions to clarify our understanding of the paper.

Keywords

Expected Value; Uniform Measure; Measure theory; Prevalence; Entropy; Sample; Linear; Superlinear; Choice Function; Bernard's Paradox; Pseudo-random

Subject

Computer Science and Mathematics, Mathematics

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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Comments (1)

Comment 1

Received: 17 March 2023
Commenter: Bharath Krishnan
Commenter's Conflict of Interests: Author

Comment: Clairifed abstract, motivation and defintion 6. Added examples to definitions 2 and 3.

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