Measure Space

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 (Ω, 𝒜, μ), composed 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,∞) ∪ {∞}.

The convention for arithmetic with ∞ in measure spaces is that:

• x + ∞ = ∞ for all x ∈ [0,∞]

• x * ∞ = ∞ for all x ∈ (0,∞]

• 0 * ∞ = 0

As for a measure, this is a map from 𝒜 to non-negative real numbers. It must satisfy two conditions:

• μ(Ø) = 0

• σ additivity: additivity for finite and countably infinite elements, but not infinite elements

  • in the finite case, requiring that each is disjoint (i.e. A_i ∩ A_j = Ø, i != j):

  • in the countable case:

It should be clear that these conditions are the reason for conditions 1 and 3 on the definition of a σ algebra.

CategoryRicottone