This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 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 nonvoid 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 Thus, there exists a neighborhood V of the origin such that ¯ V 6⊇ 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 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 convexhull ( A ) = convexhull T ( A ) for any set A . Because G is a group, G ( G A ) = G A for any set A . 4 Thus U 2 := convexhull ( G U 1 ∩ ( K 1 K 1 )) = convexhull ( G ( U 1 ∩ ( K 1 K 1 ))) ⊆ V 1 is relatively open in K 1 K 1 and satisfies G U 2 = U 2 6⊇ K 1 K 1 . By continuity, G ¯ U 2 = ¯ U 2 . Define ∞ > δ := inf { a : a > ,aU 2 ⊇ K 1 K 1 } ≥ 1 and U := δU 2 . For each < < 1 , (1 + ) U ⊇ K 1 K 1 6⊆ (1 ) ¯ U. 5 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 subcovering 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 > , 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 = ∅ . 6 Because (1 1 4 n ) ¯ U 6⊇ K 1 K 1 , we have K 2 6 = 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 ....
View
Full
Document
This note was uploaded on 02/28/2012 for the course MATH 251C taught by Professor N.r.wallach during the Winter '11 term at Colorado.
 Winter '11
 N.R.Wallach
 Logic, Transformations

Click to edit the document details