Editing Haar measure (section)

===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.