Measure Space
Contents
Introduction
To motivate a theory of measures:
"[I]n one dimension, the intervals A := [0, 1] and B := [0, 2] are in one-to-one correspondence (using the bijection x -> 2x from A to B), but of course B is twice as long as A. So one can disassemble A into an uncountable number of points and reassemble them to form a set of twice the length." (p3, An Introduction to Measure Theory, Terence Tao)
Description
A measure space is a triple of:
a sample space Ω
a σ algebra 𝒜
Note that the double (Ω, 𝒜 ) is sometimes called a measurable space.
a measure μ : 𝒜 -> [0,∞]
Note that [0,∞] is shorthand for the more formally correct [0,∞) ∪ {∞}.