R. Göbel, E. Walker's Abelian Group Theory. Proc. conf. Oberwolfach, 1981 PDF

By R. Göbel, E. Walker

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Since νv is a measure τν is σ-additive and in particular τν ∈ M+ (T ). e. τν (D) = ν (ϕ (S × D)) = ν∗ (D) and hence τν |B0 (T ) = ν∗ . 26(i)) and τν (T \ RG ) = 0 all statements of (ii) besides the uniqueness property have been shown. Clearly, any other extension τ1 of ν∗ must also be σ-finite. If RG is equipped with the induced topology the restriction ιr : RG → S × RG is continuous and ϕr := ϕ|S×RG is surjective. As B (T ) is countably generated ϕ−1 r (B) = ϕ−1 (B) ∩ (S × RG ) ∈ B (S × T ) ∩ (S × RG ) ∈ (B (S) ⊗ B (T )) ∩ (S × RG ) = B (S) ⊗ B (RG ) for all B ∈ B (H) which proves the measurability of ϕr .

51 makes intensive use of the fact that H is second countable. 52 shows that this condition is indeed indispensable. In combination with the compactness of G (-c-) this in particular ensures the existence of a disjoint decomposition of H into countably many relatively compact G-invariant Borel sets A1 , A2 , . . ∈ B (H) which has repeatedly been used to extend results on finite measures to arbitrary Borel measures. Due to (-d-) and (-f-) the pre-image ϕ−1 (A) is a cartesian product S × ι−1 (ϕ−1 (A)) ⊆ S × T for each A ∈ BG (H).

For each τ ∈ Mσ (T ) the product measure μ(S) ⊗ τ ∈ Mσ (S × T ) is invariant under the G-action on S × T . e. Φ is well-defined. 11(iv) there exist disjoint G-invariant relatively compact subsets A1 , A2 , . . ∈ BG (H) with H = j∈IN Aj . By the definition of Borel measures for any ν ∈ MG (H) we have ν∗ (ι−1 (ϕ−1 (Aj ))) = νj (Aj ) < ∞ for all j ∈ IN which proves the first assertion of (i). Let τ|B0 (T ) = ν∗ . e. 18(i)) implies μ(S) ⊗ τ = ν. 2 Definition of Property (∗) and Its Implications (Main Results) ν∗ ι−1 ϕ−1 (A) = ν(A) = μ(S) ⊗ τ S × ι−1 ϕ−1 (A) 29 = τ ι−1 ϕ−1 (A) for all A ∈ BG (H), that is τ|B0 (T ) = ν∗ which completes the proof of ϕ (i).

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