Editing Haar measure (section)

===A construction using compact subsets===

The following method of constructing Haar measure is essentially the method used by Haar and Weil.

For any subsets <math>S,T\subseteq G</math> with <math>S</math> nonempty define <math>[T:S]</math> to be the smallest number of left translates of <math>S</math> that cover <math>T</math> (so this is a non-negative integer or infinity). This is not additive on compact sets <math>K\subseteq G</math>, though it does have the property that <math>[K:U]+[L:U]=[K\cup L:U]</math> for disjoint compact sets <math>K,L\subseteq G</math> provided that <math>U</math> is a sufficiently small open neighborhood of the identity (depending on <math>K</math> and <math>L</math>).  The idea of Haar measure is to take a sort of limit of <math>[K:U]</math> as <math>U</math> becomes smaller to make it additive on all pairs of disjoint compact sets, though it first has to be normalized so that the limit is not just infinity. So fix a compact set <math>A</math> with non-empty interior (which exists as the group is locally compact) and for a compact set <math>K</math> define
:<math>\mu_A(K)=\lim_U\frac{[K:U]}{[A:U]}</math>
where the limit is taken over a suitable directed set of open neighborhoods of the identity eventually contained in any given neighborhood; the existence of a directed set such that the limit exists follows using [[Tychonoff's theorem]].

The function <math>\mu_A</math> is additive on disjoint compact subsets of <math>G</math>, which implies that it is a regular [[content (measure theory)|content]]. From a regular content one can construct a measure by first extending  <math>\mu_A</math> to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets <math>U</math>, the corresponding measure <math>\mu_A(U)</math> need not be given by the lim sup formula above. The problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure, so is not an outer measure.)