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haarmeasure - 1 Construction of Haar Measure Denition 1.1 A...

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1 Construction of Haar Measure Definition 1.1. A family G of linear transformations on a linear topological space X is said to be equicontinuous on a subset K of X if for every neighborhood V of the origin in X there is a neighborhood U of the origin such that the following condition holds if k 1 , k 2 K and k 1 - k 2 U, then G ( k 1 - k 2 ) V that is T ( k 1 - k 2 ) V for all T G . 1
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Theorem 1.2 (Kakutani). Let K be a compact, convex subset of a locally convex linear topological space X , and let G be a group of linear mappings which is equicontinuous on K and such that G ( K ) K . Then there exists a point p K such that T ( p ) = p T G Proof. By Zorn’s lemma, K contains a minimal non-void compact convex subset K 1 such that G ( K 1 ) K 1 . If K 1 contains just one point then the proof is complete. If this is not the case, the compact set K 1 - K 1 contains some point other than the origin. 2
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Thus, there exists a neighborhood V of the origin such that ¯ V K 1 - K 1 . There is a convex neighborhood V 1 of the origin such that αV 1 V for | α | ≤ 1 . By the equicontinuity of G on the set K 1 , there is a neighborhood U 1 of the origin such that if k 1 , k 2 K 1 and k 1 - k 2 U 1 then G ( k 1 - k 2 ) V 1 . 3
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Because each T G is invertible, T maps open sets to open sets (open mapping theorem) and T ( A B ) = TA TB for any sets A, B . Since T is linear, T convex-hull ( A ) = convex-hull T ( A ) for any set A . Because G is a group, G ( G A ) = G A for any set A . 4
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Thus U 2 := convex-hull ( G U 1 ( K 1 - K 1 )) = convex-hull ( G ( U 1 ( K 1 - K 1 ))) V 1 is relatively open in K 1 - K 1 and satisfies G U 2 = U 2 K 1 - K 1 . By continuity, G ¯ U 2 = ¯ U 2 . Define > δ := inf { a : a > 0 , aU 2 K 1 - K 1 } ≥ 1 and U := δU 2 . For each 0 < < 1 , (1 + ) U K 1 - K 1 (1 - ) ¯ U. 5
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The family of relatively open sets { 2 - 1 U + k } , k K 1 , is a covering of K 1 . Let { 2 - 1 U + k 1 , . . . , 2 - 1 U + k n } be a finite sub-covering and let p = ( k 1 + . . . k n ) /n . If k is any point in K 1 , then k i - k 2 - 1 U for some 1 i n . Since k i - k (1 + ) U for all i and all > 0 , we have p 1 n ( 2 - 1 U + ( n - 1) · (1 + ) U ) + k. For = 1 4( n - 1) , we have p (1 - 1 4 n ) U + k for each k K 1 . Let K 2 = K 1 k K 1 (1 - 1 4 n ) ¯ U + k = . 6
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Because (1 - 1 4 n ) ¯ U K 1 - K 1 , we have K 2 = K 1 . The closed set K 2 is clearly convex. Further since T ( a ¯ U ) a ¯ U for T G , we have T ( a ¯ U + k ) a ¯ U + Tk for all T G , k K 1 . Recalling TK 1 = K 1 for T G , we find that G K 2 K 2 , which contradicts the minimality of K 1 . 7
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Theorem 1.3 (Haar Measure). Let G be a compact group. Let C ( G ) be the space of continuous maps from G to C . Then, there is a unique linear form m : C ( G ) -→ C having the following properties: 1. m ( f ) 0 for f 0 ( m is positive). 2. m (11) = 1 ( m is normalized). 3. m ( s f ) = m ( f ) where s f is defined as the function s f ( g ) = f ( s - 1 g ) s, g G ( m is left invariant).
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