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4. Assume that the '1ft are inner • premeasures 'v't E I. Then 'If is inner • tight. Thus 'lfx for nonvoid K C I is inner • tight as well. Proof. Fix A C B in 'I, and then U E j(l) such that A,B E 'Ih. l. some D c Q \ P and M C Q 'v'M E VJl such that 'l'u(Q)- 'l'u(P)- E < inf{'l'u(M) : ME VJ'l}. l. u) :ME VJ'l}. This proves the assertion. Next we return to section 2 in order to obtain the conclusion that 'I' is • continuous at 0 . 34 have the obvious consequence which follows. 5. Assume that the 'I1 are • compact and hence the ljf1 are • continuous at Vt E I.

The resultant one-to-one correpondence α ⇔ β was then extended to arbitrary Hausdorff spaces X in Schwartz [1973], and used to define Borel-Radon measures on 43 ¨ KONIG 110 these spaces. Thus the definition of these measures came under the restriction of local finiteness, which meanwhile thanks to Berg-Christensen-Ressel [1984] could be abandoned. However, it is clear that in the present context conditions of the type of local finiteness are natural ones. In the present classical example it is nontrivial to note that α = β can happen even for locally compact X.

6. Assume that the ljf1 are • continuous at 0 Vt E I. j. m C 'I~ for some nonvoid countable K C I. Dl ljf(M) = 0. Proof. 2). I) Let I be countable, so that K =I= N. m} is> 0 and hence >some E > 0. 1). Thecasen = 1: We haveljf= 'l't x '1'{2,. {2,-··})*. m} >E. The step 1 ;£ n:::} n + 1: We have 'i'{n+l,-··} = 'l'n+l X 'i'{n+2,-··} on 'I{n+l,-··} = ('In+l X 'I{n+2,-··})*. m} > E, where M(at, ···,an)(an+ I)= M(at, .. · ,an,an+t) is obvious from the definition. This terminates the inductive choice.