The Haar Measure

In this section, we give a brief review of the measure theory which will be used in later sections. We use [R, Chapters 1 and 2] as our main resource. A σ-algebra on a set X is a collectionM of subsets of X such that ∅ ∈M, if S ∈M, then X \ S ∈ M, and if a countable collection S1, S2, . . . ∈ M, then ∪i=1Si ∈ M. That is, M is closed under complements and countable unions, and contains the empty set. A measure on a set X with σ-algebra M is a function μ : M → R≥0 ∪ {∞} such that, if {Si}i≥1 is a countable collection of pairwise dijoint elements of M, then