By Steven Dale Cutkosky
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Additional resources for An Introduction to Galois Theory [Lecture notes]
Probability measures n on M The set of invariant regular is also known to be In one-to-one convex linear correspondence with the set : of all invariant states V on Q 44 I. E. Segal (V being invariant if V (UTU-1) = v(T) for and n correspond if V v(Qk) = T E A and j(x)dn(x) for all a 6G), where k e C(M). Now is a convex set which is compact in the weak topology (recalling that the state space of a Ca-algebra with an identiti is compact). )) (Ti is a countable dense sub- d(n, p) = : 2 1 T1 = C, the metric Moreover, it T1ff-1 fn(T1) - i is easily seen to induce a topology on 37 p(T1)I identical with the weak topology.
I (Uaf)(x) = f(a(x)), f c14. Then it is easily seen that a bounded measurable complex-valued function on for all QkUa a e G. f', so that in particular `I)( Qk is the algebra of all Now it is known C = ' ^ [ Uara e G ]'. k, then UaQk = is weakly closed, and as it is easily verified that Of, = (Qk)*, m is SA and so a Ws-algebra. Plainly, is weakly closed, and it is easily seen that [ U a I a G G ]' is unitary, so that Hence (Ua)* - U -1 showing that R is a We-algebra. [ Ua+ a e G ]' Thus [ Ual a e G ] Ua is Sk.
T = Qk and 3 = I r [/M k(x)dm,, (x) if k is the set of all bounded Baire functions equation holds, it is easily seen that 7T k on in the dAA(). Now M for which this )Y is closed under bounded point- wise convergence, and as k contains all continuous functions, it consists of all bounded Baire functions. L). e. ergodic (in fact the into invariant sub-measures is valid without M). Now the ergodic invariant regular proba- bility measures on a compact space are precisely the extreme points of the set of all invariant regular probability measures on the space (the proof of this in  is for the group of reals under addition, but applies to an arbitrary group with trivial modifications).
An Introduction to Galois Theory [Lecture notes] by Steven Dale Cutkosky