Measurable Sets

Overview

The concept of measurable sets if foundational to measure theory and integration in general. A measurable set is a set that can be assigned some sort of volume. The Banach Tarski Paradox shows that it is impossible to consistently assign a measure to every subset of Euclidean space, which necesitates a theory of measurable sets.

Algebra of Sets

A class of subsets {% C %} of a set {% \Omega %} is an algebra if

• {% \Omega %} and {% \emptyset %} are in the algebra
• if A,B {% \in %} C then A {% \cap %} B {% \in %} C, A {% \cup %} B {% \in %} C, A/B {% \in %} C

C is a {% \sigma-algebra %} if {% \bigcup_{n=1}^{\infty} A_n \in C %}

Measure

A real valued set function (inputs are sets, output is a real number) on a class of sets C is called a measure if