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Vfo, ni) . . v(^, nk), . ), where M)(1, W{X, n), . ,w(xh, nk), . )» where M; (x, 72) = 2 ^rn'n (#1 (^)) y tei ^ ' ) = P 2 m' m'n (gi ($)) Pmm* (#2 (x)) U (x, m). 2 that \(gi, v)\(g2, u) = ^digi, u)- O n t n e other hand, by c) for Pmn(g), we have w(x, n) = 2 Pmn((jgig2)(x))u(x> m )Consequently, UgU^F = U^F. 1 The operators Ug form a representation of Gx in &£?. LEMMA. F2) = (Fl9 F2) for any Fu F2 G M and g G Gx. PROOF. Let F i = ( 1 , Mite, » ) , . . , _ . ^ 2 = ( 1 , ^ t e r W)» • • •» ^ 2 t e l , « l ) , .

We need a somewhat more general theorem, which can be proved similarly. 7. Let \ u X2, . , X* > 0 and X! + . . + X* < 1. Then the mapping f(x) H-* fifo — xh, . +xk-+- HM 0 . . H%h, consistent with the action of G. The meaning of the concepts and mappings introduced is the same as that explained earlier for k — 2. The mappings indicated above are consistent; namely, if i i then the mapping i, 3 is the composition of the mappings and H%. -v (g> H^... §2. Construction of the multiplicative integral of representations of G = PSL(2, R).

A representation of Gx. So far we have constructed a representation Ug of the group G° of step functions X -* G. We now show how a representation of the group Gx of continuous functions on X ->• G can be defined in terms of this representation. Namely, we claim that the representation U-g of G° (second construction) can be extended to a representation of a complete metric group containing both G° and Gx as everywhere dense subgroups. This then defines an irreducible unitary representation of Gx.

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