theorem Th4: :: MESFUNC4:4

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