On Automorphisms of Compact Groups

If m is Haar measure in G (normalized so that m(G) = 1) we consider the set function m'{E) = m{E). (E is the set of all x, # £ £ . ) Since m is a measure on G possessing all defining properties of m it follows from the uniqueness of Haar measure that ?n'(E) =m(E) for every measurable set E. In other words a is a measure preserving transformation of G; the purpose of this note is to investigate a few simple properties of a from the point of view of measure theory. We shall make use of the Pontrjagin duality theory, and, in particular, we shall need the fact that the group of automorphisms of G is essentially the same as that of the character group G*. More precisely: if to any =(#) £ G * we make correspond <£ = $(#) =4>(x), then 0 GG*, and the correspondence <£—»<£ is an automorphism of G*. The duality theory also enables us to reverse this argument: every automorphism of G* is induced in this way by a continuous automorphism of G. We recall some standard definitions from ergodic theory. A measure preserving transformation a (not necessarily an automorphism) is ergodic if the only (complex valued, measurable) solutions ƒ of the equation f =ƒ are constant almost everywhere. The transformation a is mixing if the only (complex valued, measurable) solutions ƒ of the equation f = X/, for any constant X, are constant almost everywhere. (It is true, though irrelevant, that for XT^I even a constant fails to be a solution unless it is zero.) I t is well known that the mapping