

Subject:
INTEGRATION OF NON  NEGATIVE FUNCTIONS
Category: Science > Math Asked by: madukarga 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. 

Subject:
Re: INTEGRATION OF NON  NEGATIVE FUNCTIONS
Answered By: dannidinga on 21 Nov 2002 06:12 PST Rated: 
Hi again Madukarga, 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 then g_n(x)=f(x) for all n>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 
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