Measure Theory for Beginners (Part Two): Two Languages for One System

Measurable Functions and Random Variables

Left to right: Coin Toss, Courtesy of Wikimedia Commons; The Lebesgue integral; graphical interpretation of the Lebesgue integral.

In my previous article, I promised to move on to the theory of functions and integration. On the way, I wanted to take a stop to talk about the equivalence between the axioms of probability and measure more deeply, and the kinds of functions that arise from them.

We’re picking up where we left off here.

If you haven’t already, please read the previous article to catch up to speed with where we are. This series is part of a larger series working up to Ergodic theory and Stochastic Calculus. It is written for the upper-level undergraduate with a background in some analysis or calculus, or the first-year graduate student.

Roadmap of the Article:

• Probability axioms and relation to measure
• Application of measurable functions
• Facts about measurable functions

The Kolmogorov Axioms and Measurability

The axioms of probability are formally known as the Kolmogorov axioms, and were introduced in their modern formalization by Andrei Kolmogorov in 1933. They center around a function P which assigns probabilities to some subsets of the sample…