From Haar to Lebesgue via Domain Theory

Transcription

1 From Haar to Lebesgue via Domain Theory Michael Mislove Tulane University Topology Seminar CUNY Queensboro Thursday, October 15, 2015 Joint work with Will Brian Work Supported by US NSF & US AFOSR

2 Lebesgue Measure and Unit Interval I [0, 1] R inherits Lebesgue measure: ([a, b]) = b a. I Translation invariance: (x + A) = (A) for all(borel) measurable A R and all x 2 R. I Theorem (Haar, 1933) Every locally compact group G has a unique (up to scalar constant) left-translation invariant regular Borel measure µ G called Haar measure. If G is compact, then µ G (G) = 1. Example: T ' R/Z with quotient measure from. If G is finite, then µ G is normalized counting measure.

3 The Cantor Set C 0 C 1 C 2 C 3 C 4 C = T n C n [0, 1] compact 0-dimensional, (C) =0. Theorem: C is the unique compact Hausdor 0-dimensional second countable perfect space.

4 Cantor Groups I Canonical Cantor group: C'Z 2 N is a compact group in the product topology. µ C is the product measure (µ Z2 (Z 2 ) = 1) Theorem: (Schmidt) The Cantor map C![0, 1] sends Haar measure on C = Z 2 N to Lebesgue measure. Goal: Generalize this to all group structures on C.

5 Cantor Groups I Canonical Cantor group: C'Z 2 N is a compact group in the product topology. µ C is the product measure (µ Z2 (Z 2 ) = 1) I G = Q n>1 Z n is also a Cantor group. µ G is the product measure (µ Zn (Z n ) = 1) I Z p 1 =lim n Z p n p-adic integers. x 7! x mod p : Z p n+1! Z p n. I H = Q n S(n) S(n) symmetric group on n letters. Definition: A Cantor group is a compact, 0-dimensional second countable perfect space endowed with a topological group structure.

6 Two Theorems and a Corollary I Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. I Proof: 1. G is a Stone space, so there is a basis O of clopen neighborhoods of e. If O 2O,thene O = O ) (9U 2O) U O O U O ) U 2 U O O. SoU n O. Assuming U = U 1, the subgroup H = S n Un O. 2. Given H < G clopen, H = {xhx 1 x 2 G} is compact. G H!Hby (x, K) 7! xkx 1 is continuous. K = {x xhx 1 = H} is clopen since H is, so G/K is finite. Then G/K = H is finite, so L = T x2g xhx 1 H is clopen and normal.

7 Two Theorems and a Corollary I Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. I Corollary: If G is a Cantor group, then G ' lim n G n with G n finite for each n. I Theorem: (Fedorchuk, 1991) If X ' lim i2i X i is a strict projective limit of compact spaces, then Prob(X ) ' lim i2i Prob(X i ). I Lemma: If ': G! H is a surmorphism of compact groups, then Prob(')(µ G )=µ H. Proof: A H measurable ) Prob(') µ G (ha) =µ G (' 1 (ha)) = µ G (' 1 (h)' 1 (A)) = µ G ((g ker ') ' 1 (A)) (where '(g) =h) = µ G (g (ker ' ' 1 (A)) = µ G (ker ' ' 1 (A)) = µ G (' 1 (A)) = Prob(') µ G (A).

8 Two Theorems and a Corollary I Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. I Corollary: If G is a Cantor group, then G ' lim n G n with G n finite for each n. I Theorem: (Fedorchuk, 1991) If X ' lim i2i X i is a strict projective limit of compact spaces, then Prob(X ) ' lim i2i Prob(X i ). In particular, if X = G and X i = G i are compact groups, then µ G =lim i2i µ Gi in Prob( Q i G i).

9 Two Theorems and a Corollary I Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. I Corollary: If G is a Cantor group, then G ' lim n G n with G n finite for each n. Moreover, µ G =lim n µ n,whereµ n is normalized counting measure on G n.

10 It s all about Abelian Groups I Theorem: If G =lim n G n is a Cantor group, there is a sequence (Z ki ) i>0 of cyclic groups so that H =lim n ( iapplenz ki ) has the same Haar measure as G. Proof: Let G ' lim n G n, G n < 1. Assume H n = G n with H n abelian. Define H n+1 = H n Z Gn+1 / G n.then H n+1 = G n+1, so µ Hn = µ n = µ Gn for each n, andh =lim n H n is abelian. Hence µ H =lim n µ n = µ G.

11 Combining Domain Theory and Group Theory C =lim n H n, H n = iapplen Z ki Endow H n with lexicographic order for each n; then n : H n+1! H n by n (x 1,...,x n+1 )=(x i,...,x n ) & n : H n,! H n+1 by n (x 1,...,x n )=(x i,...,x n, 0) form embedding-projection pair: n n =1 Hn and n n apple 1 Hn+1. C'bilim (H n, n, n ) is bialgebraic total order: ': K(C)! [0, 1] by '(x 1,...,x n )= P iapplen x i k 1 k i induces b': C![0, 1] monotone, Lawson continuous. µ C =lim n µ n implies for 0 apple m apple p apple k 1 k n : p k 1 k n ]) = p µ C ( b' 1 m m [ k 1 k n, k 1 k n = ([ k 1 k n, Then inner regularity implies Prob( b')(µ C )=. If C 0 =lim n G 0 n with G 0 n finite, then b' 1 m p k 1 k n ]) b' 0 : C 0 \ K(C 0 )!C\K(C) is a Borel isomorphism. strictly monotone

12 Lagniappe: Non-measurable Subgroups In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? Some known results: Every infinite compact abelian group has a non-measurable subgroup (Comfort, Raczkowski, and Trigos-Arrieta 2006) With the possible exception of metric profinite groups, every infinite compact group has a non-measurable subgroup (Hernández, Hofmann and Morris 2014) Proposition (Brian & M. 2014) Let G be an infinite compact group. 1. It is consistent with ZFC that G has a non-measurable subgroup. 2. If G is an abelian Cantor group, then G has a nonmeasurable subgroup.

13 Lagniappe: Non-measurable Subgroups Proposition (Brian & M. 2014) Let G be an infinite compact group. 1. It is consistent with ZFC that G has a non-measurable subgroup. 2. If G is an abelian Cantor group, then G has a nonmeasurable subgroup. Ad 1: By Hernández, et al., wecanassumeg is metric and profinite, so G is a Cantor group. Our results show Haar measure on G 'Cis the same as for an abelian group structure, for which b : C![0, 1] takes Haar measure to Lebesgue measure. Fact: There is a model of ZFC that admits a countable subset X [0, 1] that is not Lebesgue measurable (cf. Kechris). Then Y = b 1 (X ) Cis not Haar-measurable. H = hy i is a countable subgroup of G. ThenH is not measure 0 since then Y would be measurable, while µ G (H) > 0impliesH is open, which implies H 0.ThusH is not Haar measurable.

14 Lagniappe: Non-measurable Subgroups Proposition (Brian & M. 2014) Let G be an infinite compact group. 1. It is consistent with ZFC that G has a non-measurable subgroup. 2. If G is an abelian Cantor group, then G has a nonmeasurable subgroup. Ad 2: We first prove something stronger: 1.) If G is an infinite abelian group and p 2 G \{e}, thenthereisa maximal subgroup M < G \{p} satisfying p 2hx, Mi for all x 2 G \ M. 2.) G/M abelian =) 9 : G/M! R/Z with (p) 6= e. ker <G/M, M maximal wrt not containing p + M =) ker = M. Thus G/M ' K < R/Z. p 2hx, Mi =) pm 2hxMi (8x 2 G) =) pm =(xm) n x (9n x 2 Z). g 2 R/Z =) g has countably many roots, so G/M is countable. Choosing Q < C dense and proper and then Q < M implies M is proper, dense and has countable index. 2

15 Lagniappe: Non-measurable Subgroups In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? Some known results: Every infinite compact abelian group has a non-measurable subgroup (Comfort, Raczkowski, and Trigos-Arrieta 2006) With the possible exception of metric profinite groups, every infinite compact group has a non-measurable subgroup (Hernández, Hofmann and Morris 2014) Proposition (Brian & M. 2014) Let G be an infinite compact group. 1. It is consistent with ZFC that G has a non-measurable subgroup. 2. If G is an abelian Cantor group, then G has a nonmeasurable subgroup. A result of Hofmann and Morris implies the remaining case is C =lim n G n, G n nonabelian simple groups for each n > 0.