theorem Th69: :: MESFUNC5:69

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative & g is_simple_func_in S & g is nonnegative & ( for x being object st x in dom (f - g) holds
g . x <= f . x ) holds
( dom (f - g) = (dom f) /\ (dom g) & integral' (M,(f | (dom (f - g)))) = (integral' (M,(f - g))) + (integral' (M,(g | (dom (f - g))))) )