Editing Haar measure (section)

==Construction of Haar measure==
===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.)

===A construction using compactly supported functions===

Cartan introduced  another way of constructing Haar measure as a [[Radon measure]] (a positive linear functional on compactly supported continuous functions), which is similar to the construction above except that <math>A</math>, <math>K</math>, and <math>U</math> are positive continuous functions of compact support rather than subsets of <math>G</math>. In this case we define <math>[K:U]</math> to be the infimum of numbers <math>c_1+\cdots+c_n</math> such that <math>K(g)</math> is less than the linear combination <math>c_1 U(g_1 g)+\cdots+c_n U(g_n g)</math> of left translates of <math>U</math> for some <math>g_1,\ldots,g_n\in G</math>.
As before we define
:<math>\mu_A(K)=\lim_U\frac{[K:U]}{[A:U]}</math>.
The fact that the limit exists takes some effort to prove, though the advantage of doing this is that the proof avoids the use of the axiom of choice and also gives uniqueness of Haar measure as a by-product. The functional <math>\mu_A</math> extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure. (Note that even though the limit is linear in <math>K</math>, the individual terms <math>[K:U]</math> are not usually linear in <math>K</math>.)

===A construction using mean values of functions===

Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups. The idea is that given a function <math>f</math> on a compact group, one can find a [[convex combination]] <math display="inline">\sum a_i f(g_i g)</math> (where <math display="inline">\sum a_i=1</math>) of its left translates that differs from a constant function by at most some small number <math>\epsilon</math>. Then one shows that as <math>\epsilon</math> tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function <math>f</math>.

For groups that are locally compact but not compact this construction does not give Haar measure as the mean value of compactly supported functions is zero. However something like this does work for [[almost periodic function]]s on the group which do have a mean value, though this is not given with respect to Haar measure.

===A construction on Lie groups===

On an ''n''-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant ''n''-form. This was known before Haar's theorem.