The Haar Measure

Prerequisites

A locally compact Hausdorff group is a topological group that meets the requirements of being locally compact and the requirements of being a Hausdorff topological space.

We will let G with the operator *, described as (G,*), be the locally compact Hausdorff topological group.

Quick note on σ-algebra: σ-algebra is denoted as (X,Y) (with whatever variable replacing them) where X denotes the set, and Y is a set of subsets of X. Let X = {a, b, c, d, e}. Then a possible value for Y would be {∅, {a, b}, {c, d, e}, {a, b, c, d, e}} In this case, the σ-algebra is also the Borel σ-algebra because it is the fewest number of subsets that could contain all elements of the set X. Alternatively, Y could be {∅, {a, b}, {c, d}, {e}, {a, b, c, d, e}}, and it would not be a Borel σ-algebra because it contains one more subset than the fewest number of possible subsets within the set X.

According to MathWorld from Wolfram, "a left invariant Haar measure on G is a Borel measure μ satisfying the following conditions:

1. $ μ(xE)=μ(E) $ for every x in G and every measurable $ E {\displaystyle \subset } = G $.

2. $ μ(U) > 0 $ for every nonempty open set U subset= G.

3. $ μ(K) < \infty $ for every compact set $ K {\displaystyle \subset } = G $ " (Moslehian).