theorem :: MESFUNC5:32

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st f is_simple_func_in S & dom f = {} holds
ex F being Finite_Sep_Sequence of S ex a, x being FinSequence of ExtREAL st
( F,a are_Re-presentation_of f & a . 1 = 0 & ( for n being Nat st 2 <= n & n in dom a holds
( 0 < a . n & a . n < +infty ) ) & dom x = dom F & ( for n being Nat st n in dom x holds
x . n = (a . n) * ((M * F) . n) ) & Sum x = 0 )