f and g are non negative measurable functions.

(X, ) is the measure space

for A

Prove

so i have by monotone conv thm =

and there exists a increasing seq of simple functions s.t.

and

now =

if each takes on finite number of values k dependent on n and each takes on finite number of values l dependent on m

and and let

=

=

this following part is where i am kind of unsure if i did it correctly

=

let

so if i is fixed and j runs from 1 to l, then it becomes

the limiting operations dont necessarily commute i.e lettin n to go infinity and then m to infinity is not the same as m going to infinity and then n going to infinity as the sum need not converge absolutely. any hints or help would be appreciated