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Q: INTEGRATION OF NON - NEGATIVE FUNCTIONS ( Answered ,   0 Comments )
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 Subject: INTEGRATION OF NON - NEGATIVE FUNCTIONS Category: Science > Math Asked by: madukar-ga List Price: \$5.00 Posted: 18 Nov 2002 09:29 PST Expires: 18 Dec 2002 09:29 PST Question ID: 109961
 ```If f belongs to L+(the space of all measurable functions from X to [0,infinity] ) and integral f < infinity, for every eplison >0 there exists E belongs to Script M such that mu(E)< infinity and integralto the base E f >(integral f) - epsilon.```
 ```Hi again Madukar-ga, For n=1,2,3,..., define the set E_n = {x in X : f(x) < n} Define the function g_n = f * indicator (=characteristic) function of E_n. g_n is a sequence of positive functions that converges pointwise to f, since for any x in X, if f(x)n_0. Also all the g_n are bounded by f, which is an integrable function. Therefore by the dominated convergence theorem, we know that integral(g_n)-->integral(f) as n->infinity. In particular, for some value of n we will have that integral(g_n) > integral(f)-epsilon. But integral(g_n) = integral(f) over the set E_n. So E_n is the E that you asked for. Hope everything's clear, if not I'm here for you... Regards, dannidin```