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
{% \nu(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} \nu(A_n) %}