On a Review of the Lebesgue Integrals with Application to Probability Theory | Asian Research Journa

The study of Lebesgue measure theory is motivated by two fundamental motivations: a) the desire to measure the length of any subset of the real line; and b) the need to have a theory of the integral in which the syllogism is preserved.

Holds in the broadest sense conceivable. Both of these desires turn out to be too unclear. In fact, (a) is illogical. We must confine our focus to a certain class of sets in order to establish a practical and usable theory of measuring sets. In terms of (b), we can absolutely develop an integral theory in which (I) is simple and natural. However, there is no such thing as a "optimal" hypothesis.

Both of the above concerns are well-addressed by the Lebesgue integral. We'll devote a few pages to this project to provide a quick overview of the key concepts. We won't be able to prove all of the conclusions, but we will be able to articulate them clearly and present some illustrative instances. In this paper, we will look at how some of the Lebesgue measure's most fundamental discoveries can be applied to probability theory.

It's worth noting that the concept of length that we'll build here is referred to as a "measure." Probability, on the other hand, stretches back to the days of B. Pascal (1633-1662) and even before that, when gamblers wanted to predict the outcome of specific bets. The issue lacked spatial development and was riddled with contradictions and ambiguities. The subject could not be put on a firm foundation until 1933, when A.N. Komogorov (1903-1987) realised that measure theory was the suitable language for formulating probabilistic assertions.

Please see the link :- https://globalpresshub.com/index.php/ARJOCS/article/view/793