Haar Measure - Preliminaries

Preliminaries

Let (G,.) be a locally compact Hausdorff topological group. In this article, the σ-algebra generated by all compact subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S as follows:

• Left translate:

• Right translate:

Left and right translates map Borel sets into Borel sets.

A measure μ on the Borel subsets of G is called left-translation-invariant if for all Borel subsets S of G and all g in G one has

A similar definition is made for right translation invariance.