By Benno Fuchssteiner

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**Example text**

Valued functions li- 0 with z - hi = 1 -such that h i f i ( x ) -for a l l x E X . - For every f i n i t e nonempty subset X c X w e have -- x x E Proof: ( i ) * ( i i ) : By adding u p the inequality in ( i ) for the we see that t- ~ ( ii s) dominated by an arithmetic mean of the corres- ;EX ponding sums for the f l metic mean of numbers. n to fn . 211 (construction below) we know that there exists a free abelian semigroup X* 3 X with neutral element which has the property t h a t any map T from X into an abelian semigroup S with neutral element can be uniquely extended t o an additive function from X* into S .

RxR A l l what remains t o prove i s t h e additivity o f a- B E Z and v a r i a b l e . e. n An nEN (3) = v i n the f i r s t An E X w i t h An + 0 0 and A1 2 A2 2 ...? An 2 . . ) . A l l we have t o show i s : l i m v(An,B) n-m = 0. F o r t h i s we s t a r t by s t a t i n g two h e l p f u l convergence p r o p e r t i e s . F i r s t we consider B, E II such t h a t 0 5 v(An,Bn) Ann B, 5 T ( A x~ (Bn Hence (4) l i m v(An,Bn) = = 0 n-co 0 . Then n CA,)) u s i n g ( b ) and ( c ) we g e t : 5 T ( A ~x Q ) .

T h i s shows t h a t (**) i s necessary and s u f f i c i e n t f o r e q u a l i t y i n (*). 4 When i s a F u n c t i o n Dominated by an A r i t h m e t i c Mean o f o t h e r Functions? I n t h i s subsection we prove a r e s u l t o f a somewhat c o m b i n a t o r i c a l nature, showing t h a t t h e F i n i t e Decomposition Theorem can be extended t o t h e most general s i t u a t i o n . ,f n ? Theorem: (i) The following __ are There are n cp(x) I (ii) cp i =1 cp . , and f l y . , f n on X pointwise dominated by an arithmetic mean equivalent: .