By W. & H. Knapp Grobner
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Extra resources for Contributions to the method of Lie series
Example text
E. E ⊆ B ∪ N = B + (B c ∩ N ), ∞ and hence E ∈ B (H)ν . This implies the inclusions B (H)ν ⊇ j=1 B (H)νj ⊇ B (H)F , and hence each G-invariant Borel measure can be uniquely extended to B (H)F (cf. 20(iii)). In particular, B (H)F = ν∈MG (H) B (H)ν . We denote this extension briefly with νv instead of νv|B(H)F . 31. (i) Suppose that (Ω, A) is a measurable space and that E is a Polish space. Further, let πΩ : E × Ω → Ω denote the projection onto the second component, and suppose that the set N ⊆ E × Ω fulfils the following conditions: (α) For each ω ∈ Ω the set N (ω) := {e ∈ E | (e, ω) ∈ N } ⊆ E is closed.
By assumption, ϕ◦ι◦ψ(hj ) ∈ EG ({hj }) for all hj ∈ H, and hence m (ϕ ◦ ι) ◦ ψ m (h1 , . . , hm ) ∈ EG ({h1 }) × · · · × EG ({hm }) for all h1 , . . , hm ∈ H. As Ψ ∗ is constant on each G-orbit EG ({h1 })×· · ·×EG ({hm }) we conclude Ψ ∗ = Ψ ∗∗ ◦ ψ m . 44. 45. (i) The Gm -orbit of (h1 , . . , hm ) ∈ H m is given by the product set EG ({h1 }) × · · · × EG ({hm }). 44 need not be constant on the G-orbits of H. If sel: H → H denotes a measurable selection (cf. 32(iii)) then ψ := ψ ◦ sel induces a sufficient statistic which additionally is constant m m on each Gm -orbit.
M ∈ M1G (H). 16) 42 2 Main Theorems where τj ∈ Φ−1 (νj ) denotes the unique pre-image of νj with τj (RG ) = 1. Proof. Let C1 , . . , Cm ∈ B (T ) and η := ν1 ⊗ · · · ⊗ νm ∈ M1G (H)m . From the definition of ψ m we obtain (ψ m )−1 (T × · · · × T × Ck × T × · · · × T ) = H ×· · ·×H ×ψ −1 (Ck )×H ×· · ·×H with ψ −1 (Ck ) ∈ B (H)F . Hence there exist disjoint subsets Ak , Bk ∈ B (H) with νk (Bk ) = 0 and ψ −1 (Ck ) ⊆ Ak + Bk . e. (ψ m )−1 (T × · · · × Ck × · · · × T ) ∈ B (H m )η . As η was arbitrary (ψ m )−1 (T × · · · × Ck × · · · × T ) ∈ CF .