Probability Theory and the Lebesgue Integral

Summary

Editors' Note: After his undergraduate years at Stetson University, Botts went to the University of Virginia where he received his PhD in 1942. After a short stay at the University of Delaware, he took a faculty position at the University of Virginia, where he remained until he became Executive Director of the Conference Board of the Mathematical Sciences, a position he held from 1968 till 1982. He is now retired and living in Arlington, Virginia.

Random phenomena

Probability theory is the study of mathematical models for random phenomena. A random phenomenon is an empirical phenomenon whose observation under given circumstances leads to various different outcomes. When a coin is tossed, either heads or tails comes up; but on a given toss we can't predict which. When a die is tossed, one of the six numbered faces comes up; but again we can't predict which. In such simple random phenomena the various possible outcomes appear to occur with what is called “statistical regularity”. This means that the relative frequencies of occurrence of the various possible outcomes appear to approach definite limiting values as the number of independent trials of the phenomenon increases indefinitely.

If we are trying to set up an abstract mathematical model for a random phenomenon, one of the features of the model should surely be a set E whose elements correspond to the various possible outcomes of this random phenomenon.