By Mac Lane, Birkhoff (ALLOUCH, MEZARD, VAILLANT, WEIL)
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Extra resources for Algebre, solutions developpees des exercices, 2eme partie, algebre lineaire [Algebra]
Assume that for each I i ~ I ~ is compact p i/~i - approximable where Pi(B): = = P(ZliI(B)) for each B a ~ compact P/~ - approximable. Proof: is a p-content, Pi/~ too, for each i E I. ) Because P / Z I i I ( ~ ) is a p-content Pi/~ and each i ~ I. T h e n ~ is P / Z i i l ( ~ ) system Zlil(~i). 2). 8) Let X = ~ ( X i and assume that over I each space X i a tight topology is defined. L e t ~ 1 be the Borel - algebra on X i and let P/A be a p-content such that P / Z ~ i } ( ~ ) is a measure for each i E I.
Then, J U Let A I , A 2 ~ JU ~j. Then, A E ~ j o , whence A C ~ j o j~j. AI~ J[J1' A 2 ~ J 2 . As J is directed, there exists Jo such that Ji < Jo' i = 1,2. Hence, A i a ~ J i c J%Jo for i = 1,2. We have A I N A 2 ~ J o ~ U~j. J Let I be an arbitrary index set and ~o the system of all finite subsets of I. 2) If (~i)i~i is a family of algebras on X, then ~(UJ[i ) -- U (UJ%i)"~. I ~o Io Proof: From (U J~i )~v ~ Io we immediately obtain a(U3li) for each subset I o I I U (UJ%i)"u ~ a(UJ%i). 1) yields that algebra.
6) imply that there exists a compact system ~i ~ o(~i ) P/o(~i ) - approximating o(~i) for each i a I. 1), version 2, for I o and ~ o instead of I and ~* and w i t h ~ i = a(~i) , (i) yield that a ( U ~ i) is I~ v~ - approximable by the compact system (U ~i ) . P/~o I o Hence, a(U S i) is P/~ - approximable by the compact system Io (U C i)u ~ , for each I o G ~o" Therefore, ~ O ~ o a ( U ~ S i) is I I~ P/~ - approximable by the compact system (U ~i) U~. 4) I this implies that P is a-additive on J~o.