Editing Haar measure (section)

==Preliminaries==
Let <math> (G, \cdot)</math> be a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]].  The  [[Sigma-algebra|<math>\sigma</math>-algebra]] generated by all open subsets of <math>G</math> is called the [[Borel algebra]]. An element of the Borel algebra is called a [[Borel set]].  If <math>g</math> is an element of <math>G</math> and <math>S</math> is a subset of <math>G</math>, then we define the left and right [[Coset|translates]] of <math>S</math> by ''g'' as follows:
* Left translate: <math display="block"> g S = \{g\cdot s\,:\,s \in S\}.</math>
* Right translate: <math display="block"> S g = \{s\cdot g\,:\,s \in S\}.</math>

Left and right translates map Borel sets onto Borel sets.

A measure <math>\mu</math> on the Borel subsets of <math>G</math> is called ''left-translation-invariant'' if for all Borel subsets <math>S\subseteq G</math> and all <math>g\in G</math> one has
:<math> \mu(g S) = \mu(S). </math>
A measure <math>\mu</math> on the Borel subsets of <math>G</math> is called ''right-translation-invariant'' if for all Borel subsets <math>S\subseteq G</math> and all <math>g\in G</math> one has
:<math> \mu(S g) = \mu(S). </math>