theorem :: MESFUNC8:30

for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being with_the_same_dom Functional_Sequence of X,ExtREAL
for g being PartFunc of X,ExtREAL
for E being Element of S st dom (f . 0) = E & ( for n being Nat holds f . n is_measurable_on E ) & M . E < +infty & ( for n being Nat ex L being Element of S st
( L c= E & M . (E \ L) = 0 & ( for x being Element of X st x in L holds
|.((f . n) . x).| < +infty ) ) ) & ex G being Element of S st
( G c= E & M . (E \ G) = 0 & ( for x being Element of X st x in E holds
f # x is convergent_to_finite_number ) & dom g = E & ( for x being Element of X st x in G holds
g . x = lim (f # x) ) ) holds
for e being Real st 0 < e holds
ex F being Element of S st
( F c= E & M . (E \ F) <= e & ( for p being Real st 0 < p holds
ex N being Nat st
for n being Nat st N < n holds
for x being Element of X st x in F holds
|.(((f . n) . x) - (g . x)).| < p ) )