Introduction to Liouville Numbers

2023

Preprint

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.

“…We need additional examples, where is not the counting measure. Perhaps one example of { } ∈ℕ (where is the Liouville numbers [6]) is:…”

Section: Defining the Most Natural Extension Of The Expectedmentioning

confidence: 99%

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

2023

Preprint

“…(4) Can either equations 3.4.1, 3.4.2 and 3.4.3 (when is the set of all Liouville numbers [6] and = id ) give a nite value? What would the value be?…”

Section: Defining the Most Natural Extension Of The Expected Valuementioning

confidence: 99%

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

2023

Preprint

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.

“…‡ [ ] that fully answers the question in §2? (4) Can either ̈ [ ], † [ ], or ‡ [ ] from equations 3.3.1, 3.3.2 and 3.3.3 respectively (when is the set of allLiouville numbers[6] and = id ) give a nite value? What would the value be?…”

mentioning

confidence: 99%

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

2023

Preprint

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on the Caratheodory extension to a larger class of functions, since the set of all functions with infinite or undefined expected values may form a prevalent subset of the set of all measurable functions. Before we get to the specific problem (or main question) of the paper, we will outline some preliminary definitions. We then will define a precise main question that will attempt to offer a unique solution and we'll offer a partial solution to the question. Along the way, we will ask a series of questions that will clarify our understanding of the paper.