Cantor Groups, Haar Measure and Lebesgue Measure on [0, 1]

Size: px

Start display at page:

Download "Cantor Groups, Haar Measure and Lebesgue Measure on [0, 1]"

Rachel Hudson
5 years ago
Views:

1 Cantor Groups, Haar Measure and Lebesgue Measure on [0, 1] Michael Mislove Tulane University Domains XI Paris Tuesday, 9 September 2014 Joint work with Will Brian Work Supported by US NSF & US AFOSR

2 Lebesgue Measure and Unit Interval [0, 1] R inherits Lebesgue measure: λ([a, b]) = b a. Translation invariance: λ(a + x) = λ(a) for all (Borel) measurable A R and all x R.

3 Lebesgue Measure and Unit Interval [0, 1] R inherits Lebesgue measure: λ([a, b]) = b a. Translation invariance: λ(a + x) = λ(a) for all (Borel) measurable A R and all x R. 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 λ. If G is finite, then µ G is normalized counting measure.

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

5 Cantor Groups 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)

6 Cantor Groups 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.

7 Cantor Groups 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) Definition: A Cantor group is a compact 0-dimensional second countable perfect space endowed with a topological group structure. G = n>1 Z n is also a Cantor group. µ G is the product measure (µ Zn (Z n ) = 1) Z p = lim n Z p n p-adic integers. H = n S(n), where each S(n) is the symmetric group on n letters.

8 Two Theorems and a Corollary Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. Proof: 1. G is a Stone space, so there is a basis O of clopen neighborhoods of e. If O O, then e O = O ( U O) U O O U O U 2 U O O. So U n O. Assuming U = U 1, the subgroup H = n Un O. 2. Given H < G clopen, H = {xhx 1 x G} is compact. G H H by (x, K) 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 = x G xhx 1 H is clopen and normal.

9 Two Theorems and a Corollary Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. Corollary: If G is a Cantor group, then G lim n G n with G n finite for each n.

10 Two Theorems and a Corollary Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. Corollary: If G is a Cantor group, then G lim n G n with G n finite for each n. Theorem: (Fedorchuk, 1991) If X lim i I X i is a strict projective limit of compact spaces, then Prob(X ) lim i I Prob(X i ).

11 Two Theorems and a Corollary Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. Corollary: If G is a Cantor group, then G lim n G n with G n finite for each n. Theorem: (Fedorchuk, 1991) If X lim i I X i is a strict projective limit of compact spaces, then Prob(X ) lim i I Prob(X i ). Lemma: If ϕ: G H is a surmorphism of compact groups, then Prob(ϕ)(µ G ) = µ H.

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

13 Two Theorems and a Corollary Theorem: If G is a compact 0-dimensional group, then G has a neighborhood basis at the identity of clopen normal subgroups. 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, where µ n is normalized counting measure on G n.

14 It s all about Abelian Groups 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 ( i n Z ki ) has the same Haar measure as G. Proof: Let G lim n G n, G n <. Assume H n = G n with H n abelian. If φ n : G n+1 G n is surjective, then G n+1 / ker φ n G n. So G n+1 = G n ker φ n. Define H n+1 = H n Z ker φn. Then H n+1 = G n+1, so µ Hn = µ n = µ Gn Hence µ H = lim n µ n = µ G. for each n, and H = lim n H n is abelian.

15 Combining Domain Theory and Group Theory C = lim n H n, H n = i n Z ki Endow H n with lexicographic order for each n; then π n : H n+1 H n by π n (x 1,..., x n+1 ) = (x 1,..., x n ) & ι n : H n H n+1 by ι n (x 1,..., x n ) = (x 1,..., x n, 0) form embedding-projection pair.

16 Combining Domain Theory and Group Theory C = lim n H n, H n = i n Z ki Endow H n with lexicographic order for each n; then π n : H n+1 H n by π n (x 1,..., x n+1 ) = (x 1,..., x n ) & ι n : H n H n+1 by ι n (x 1,..., x n ) = (x 1,..., x n, 0) form embedding-projection pair. C bilim (H n, π n, ι n ) is bialgebraic chain: C totally ordered, has all sups and infs K(C) = n {(x 1,..., x n, 0,...) (x 1,..., x n ) H n } K(C op ) = {sup ( k \ {k}) k K(C)}

17 Combining Domain Theory and Group Theory C = lim n H n, H n = i n Z ki Endow H n with lexicographic order for each n; then π n : H n+1 H n by π n (x 1,..., x n+1 ) = (x 1,..., x n ) & ι n : H n H n+1 by ι n (x 1,..., x n ) = (x 1,..., x n, 0) form embedding-projection pair. C bilim (H n, π n, ι n ) is bialgebraic chain: ϕ: K(C) [0, 1] by ϕ(x 1,..., x n, 0, 0,...) = i n monotone induces ϕ: C [0, 1] monotone, Lawson continuous. Direct calculation shows: x i k 1 k i strictly µ C ( ϕ 1 (a, b)) = λ((a, b)) for a b [0, 1]; i.e., Prob( ϕ)(µ C ) = λ.

18 Alternative Proof 1. Cantor Tree: CT Σ = Σ Σ ω, Σ = {0, 1} s t ( u) su = t. Then Max CT C.

19 Alternative Proof 1. Cantor Tree: CT Σ = Σ Σ ω, Σ = {0, 1} s t ( u) su = t. Then Max CT C. 2. Interval domain: Int([0, 1]) = ({[a, b] 0 a b 1}, ) φ: C [0, 1] extends to Φ: CT Int([0, 1]) Scott continuous. Then Prob(Φ): Prob(CT ) Prob(Int([0, 1])), so λ = Prob(Φ)(µ C ) = lim Prob(Φ)(µ n ), where Prob(Φ)(µ n ) = 1 i 2 n 1 2 n δ [ i 1 2 n, i 2 n ]

20 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν.

21 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 1 : X X [0, 1] N & X is dense G δ in X ; Prob(X ) Prob(X ).

22 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 1 : X X [0, 1] N & X is dense G δ in X ; Prob(X ) Prob(X ). 2 : X is a continuous image of C: ψ : C X.

23 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 1 : X X [0, 1] N & X is dense G δ in X ; Prob(X ) Prob(X ). 2 : X is a continuous image of C: ψ : C X. 3 : Prob(ψ): Prob(C) Prob(X ).

24 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 1 : X X [0, 1] N & X is dense G δ in X ; Prob(X ) Prob(X ). 2 : X is a continuous image of C: ψ : C X. 3 : Prob(ψ): Prob(C) Prob(X ). 4 : So it suffices to show every ν Prob(C) satisfies the Corollary.

25 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 4 : So it suffices to show every ν Prob(C) satisfies the Corollary. ν Prob(C) qua valuation on Σ(C) preserves all suprema, so F ν : C [0, 1] by F ν (x) = ν( x) preserves all infima F ν is the cdf of ν.

26 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 4 : So it suffices to show every ν Prob(C) satisfies the Corollary. ν Prob(C) qua valuation on Σ(C) preserves all suprema, so F ν : C [0, 1] by F ν (x) = ν( x) preserves all infima F ν is the cdf of ν. F ν has a lower adjoint χ ν : [0, 1] C preserving all suprema; χ ν (r) x iff r F ν (x).

27 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 4 : So it suffices to show every ν Prob(C) satisfies the Corollary. ν Prob(C) qua valuation on Σ(C) preserves all suprema, so F ν : C [0, 1] by F ν (x) = ν( x) preserves all infima F ν is the cdf of ν. F ν has a lower adjoint χ ν : [0, 1] C preserving all suprema; χ ν (r) x iff r F ν (x). So λ({r χ ν (r) x}) = λ({r r F ν (x)}) = λ([0, F ν (x)]) = F ν (x). Now φ: C [0, 1] has an upper adjoint φ : [0, 1] C.

28 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: 4 : So it suffices to show every ν Prob(C) satisfies the Corollary. ν Prob(C) qua valuation on Σ(C) preserves all suprema, so F ν : C [0, 1] by F ν (x) = ν( x) preserves all infima F ν is the cdf of ν. F ν has a lower adjoint χ ν : [0, 1] C preserving all suprema; χ ν (r) x iff r F ν (x). So λ({r χ ν (r) x}) = λ({r r F ν (x)}) = λ([0, F ν (x)]) = F ν (x). Now φ: C [0, 1] has an upper adjoint φ : [0, 1] C. Then ξ ν = φ χ ν : C C is lower adjoint to φ F ν : C C, so it preserves all sups and gives the desired result.

29 Skorohod s Theorem Theorem: (Skorohod) If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : [0, 1] X satisfying Prob(ξ ν )(λ) = ν; i.e., ν(a) = λ(ξν 1 (A)). Corollary: If ν is a Borel probability measure on a Polish space X, then there is a measurable map ξ ν : C X satisfying Prob(ξ ν )(µ C ) = ν. Proof using Domain Theory: Then ξ ν = φ χ ν : C C is lower adjoint to φ F ν : C C, so it preserves all sups and gives the desired result. Remark: Notice that, in the case of ν Prob(C), we have proved more than claimed, since ξ ν preserves all suprema, and so it is Scott continuous, rather than being just measurable.

From Haar to Lebesgue via Domain Theory

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 Lebesgue