By Steven Dale Cutkosky

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**Additional resources for An Introduction to Galois Theory [Lecture notes]**

**Sample text**

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 [9] 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

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