Preprint Concept Paper Version 8 Preserved in Portico This version is not peer-reviewed

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

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.v8 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.v8

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 forms a prevalent subset of the set of all measurable functions, which means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, we will outline some preliminary definitions. We'll then define the specific problem (along with a partial solution in section 3) to visualize the complete solution. Along the way, we will ask a series of questions to clarify our understanding of the paper.

Keywords

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

Subject

Engineering, Automotive Engineering

Comments (1)

Comment 1
Received: 30 March 2023
Commenter: Bharath Krishnan
Commenter's Conflict of Interests: Author
Comment: Added examples to each definition (despite lack of mathematical proofs) and corrected definition 6, (3).
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