By A Seidenberg
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E. STx for x E :D(T) n y-l (:D(S) ). It can happen that :D(ST) = 0. = S(Tx) 42 XIII Ergodic Transformation Groups Let A 1 be an array for Jf1 with order 2m. Consider the partial transformations 0 _:s a,b <2m, Tj(b,a) = U1(0,b)Tj U1(a,O), each of which belongs to [~z 1 (OJ 1 _:s j _:sf; J. In the natural sense, we have Tj = LUI(b,O)Tj(b,a)UI(O,a). b,a For each a E Zzm, let /La(B) = ~-t(UI (a, O)B) for B(Z1 (0) ). La(B) < 8/4m. (ii) to ~z 1 (O), we find a subgroupoid Jf~ of bounded finite type I with Jf~(O) = Z 1(0) such that £/2 I Tj(b, a) E [Jf2 ], 1 :S j :Sf, 0 :Sa< 2m.
Let U2,j(i) be a partial transformation of fj. such that and We now extend the system {U2,j (i)} as follows: U2,j(sqj + r) = Ul,j(S, O)Uz,j(r), 0 :S s < nj. 0 :S r < qj. Since JL(Z2,j (0)) = 1j2n for j = 1, 2, ... , k, { Z 2,j (0) : 1 :s j :S k} are mutually equivalent. Let Vj be a partial transformation of fj. such that V1 = idz 2• 1(OJ . With N L~=l nj, we set = U2(n1 + nz + · · · + nj + i) = U2,j+t(i)Vj+l. Zz(a) = U2(a)Zz,t(O), 0 :S i < nj+t; a= 0, 1, ... , N- 1. It then follows that U Z2(a) = X - R.
Therefore, it associates a covariant system {~ = /RK(X, G, /1), R, e}. 23. L} be an ergodic standard a-finite G-measure space with G a countable discrete group. A G, R, 8} . § 2 Krieger's Construction and Orbit Structure 29 PROOF: As e and the action of G on A commute, e gives rise to a one parameter automorphism group { et : t E R } on 5?. such that 5?. >4o R ::::::: :R K ( G x R, X, /1). To avoid possible confusion, let us denote the action of G on :B = L 00 (X, JL) by a and that on A = L oo (X, /1) by a whilst we keep writing e for the actions of R on A as well as on L 00 (R~).