By Y Yamasaki

This e-book is predicated on lectures given at Yale and Kyoto Universities and offers a self-contained special exposition of the subsequent topics: 1) the development of endless dimensional measures, 2) Invariance and quasi-invariance of measures less than translations. This e-book furnishes an incredible instrument for the research of actual structures with countless levels of freedom (such as box conception, statistical physics and box dynamics) by means of delivering fabric at the foundations of those difficulties.

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30 Remark 1« A more careful discussion shows that the conditions (1) and (2) in Th. 2 can be replaced by: that (1 )* (Y^, B^-measurable set separated, or ( 2 ) * p^^ Remark 2. 1. §8 . is a - Y^. can be extended to a measure is not always thick in v( ff A, )=1 k=l ^ there exists a thick v. In this case, v on A counter example is v exists but we have so that the diagonal is not thick. 1. Let surable spaces. (X) are measurable. algebra in which all projections p (X), X->X' namely 1B= 1B( U ( 8. 1) XeA Let product L be a subset of (X- j ^ t ) ^ ^ Xj^, and from Let Ju Let X^, We shall define the partial In a similar way: where X^^^ (Xe L) .

Before the proof of the converse, we shall remark two facts. As easily checked (or as well known), implies that its section B ’. Secondly, a-algebras X E(x)={x’ é X*| (x,x’) ^ E } E ^ B x B ’ ® and E eB x B * belongs to Implies that for some a-generated cB*, E already belongs to as proved in the next lemma. Suppose that G((f ) 6 l8 x © » . 3-[. B- , therefore Thus (X’,T3^) is a-separated, is so. i7l be a family of subsets of X, then we have Bca) =. U B(Æ'), oc where 01' Proof. = OC k=l Evidently we We shall show that is a a-algebra.

If Luzin (resp. Suslin) spaces, then their disjoint countable union is a Luzin (resp. Suslin) space in the sum topology, because: f=fj^ Z^. on If each ^k”^k^^k^ Z^ X=f( 0 Z ^ ) imply where is a complete separable metric space, then their disjoint union is so in the sum topology, and if each f^ is continuous, then f is so in the sum topologi es . Theorem 13»2. In a Luzin (resp. Suslin) space (X,t ), every Borel set is a Luzin (resp. Suslin) set. Proof. in Oi Let be the family of all Luzin (resp.