# A Theory of Semigroup Valued Measures by M. Sion

By M. Sion

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LEMMA. Let L be a G - m o d u l e , A i t s Let A1,... A set of w e i g h t s . be the E - c o n n e c t e d components of A. ~ L). Put L I• = lEA. l Then (i) some Each indecqmposable G-submodule of L is c o n t a i n e d in L.. l (ii) The L. are G - s u b m o d u l e s . I (iii) - - L : ® L.. i l PROOF. 2). As the X+j~ are d i r e c t l y ~-connected to ~, we see that x ( t ) . L i C L i. (iii) is obvious. ). i i of L. ). 14. NOTATION. Suppose LM,/M has a composition .. 5). /Vk, series LM,/M = L 1 D L 2 D ..

Z*~,6 = such that ~+8 is on ~+8. It is c l e a r : 0). Hence while ~+8 we that consider = Y+~ is n. 6, suppose (iii)). = < ~ + 8 , 8 > = 2. So = i, a contradiction. (~,~) : ! 1. X*Y-8' image G2, > (8,8), Ny_8, ~ : ! 1. For the = -N _~,6 for X6, an 7) Y into only ~ 1, < y + 6 , B > = -3, = 0 shows that to E Z, to get (~,8) N6,~_~ (ker n. distinct = (6,~). ~ 1, <~,8> Now s(~,B) (~,8) (8,8) y-6 E ~. Then 7 is c o n t a i n e d that independent depends are respect and we h a v e suppose Z* ~,8,y,6 with = (Y,Y) We h a v e (~ e ~), to some (We r e q u i r e that degenerate (~,~) come inverse [X * , X ~] , if ~ ,B are respect In this sums ker .

2), under Ad. 2. ~ of ~* (see x Ad(y) on K* component [ 1 ] , Cor. ~y discussed [p]-structure connected that 0 ~, h e n c e subalgebra (see surjective. such So it is the [ p ] - s t r u c t u r e R u of G this (~). ~ is s u r j e e t i v e , Then ~. (I). 2). 5)). Lie a l g e b r a So (ker unipotent In fact ~)0, radical we have: LEMMA. R = ker PROOF. G * / R u is c o n n e c t e d , 4: G * / R u ~ G. The has the root same system. 5. ~ Let Now ~ ~. (See as G, the there G ~ be a c o n n e c t e d Suppose that is an = ker the unipotent = [ ~ / r--U is the 23, inverse Lie a l g e b r a for Th@or~me algebraic of G ~ The semisimple (G* ~ G*/Ru) [p]-structure radical homomorphism connected.