theorem Th16: :: MESFUN10:16

for X being non empty set
for F being with_the_same_dom Functional_Sequence of X,ExtREAL
for S being SigmaField of X
for M being sigma_Measure of S
for E being Element of S
for P being PartFunc of X,ExtREAL st E = dom (F . 0) & E = dom P & ( for n being Nat holds F . n is E -measurable ) & P is_integrable_on M & ( for x being Element of X
for n being Nat st x in E holds
|.(F . n).| . x <= P . x ) holds
( ( for n being Nat holds |.(F . n).| is_integrable_on M ) & |.(lim_inf F).| is_integrable_on M & |.(lim_sup F).| is_integrable_on M )