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By Nastasescu C.

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And J( XI) ~ J+(YI) - Val ,i,cp(i) , i (n = 2,3 , ... ) . 3 there exists a bounded sequence (ai,j) i,j C G , ai,j \. 3 EXTENSION OF INTEGRALS for every ip : 51 N -t N. ; E L , there are bn,i,j \. there are V n E A -, V n ~ Xn a (j -t (0 ) such th at to every ip : N -t N Put bO ,i,j = Vn J( x n) - J( Xj). p : N -t N . 2)) v :s; x :s; u and J+(u) - Va i,cp(i) ~ w ~ J-(v) + Vbi,cp(i) . i Therefore x EL and J( x) = w = o Vn J( x n). 14 Corollary. Let Xn E L , Xn ~ x EX, (J( xn))n be bounded . Th en X l - x EL and n Proof.

Hence condition 3 is satisfi ed , to o. 1) for all X E Y. Since m(x) ~ f( x) is evident , we have to pr ove f (x ) ~ 2m(x) only. Let x = V7=1xi . Put a = I: 7=1 f (xd . 2) f (x) ~ 2Lf(Xi) = 2a. i=l If a < 00 , th ere ar e two possibilities: 1. f (xd < a/2 for all i. 2. There is i such th at f( xd ~ a/2. In the first case choose the maximal k such that k-1 I: f( Xi) < ~ . i=l Since n Lf(xd i=l we obtain n ~~ , k L f( Xi ) = a - L f( Xi) i=k+1 i=l Because of t he inductiv e assumption we have k- 1 ) f ( i~l Xi ~ ~.

X E A , X n ~ x, then J(x) = lim J( x n). 3 Examples. 1. A finit e measure on a ring . This was the first of our examples pr eceding our general definition. The lattice X consists of all subs ets of a given set n, A ~ B if and only if A c B . Lattice operations coinside with set-theoretical operations A V B = AU B , A 1\ B = An B. 2. An integral on continuous funct ions. This was the second of our motivating examples. Here X consists of all real functions continuous on a compact interval [a ,b] .

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The Category of Graded Rings by Nastasescu C.


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