Thread: Lebesgue Integral

Let the sequence of integrable functions be defined on the interval in which on .
Prove that there is no integrable function that dominates the sequence.

My work:
Suppose there exist a function that dominates the sequence.
Hence,





I was stuck here.
Can anyone help me to proceed?
Or if this is not the way to prove,can anyone please show me the correct way?
Thanks.

If there existed such a dominating function, you could apply Lebesgue Dominated Convergence theorem, but

pointwise (even uniformly!)

and

for each

Hence we can't interchange limit and integral, a contradiction with having the condictions to apply Lebesgue theorem.