The Fatou-Lebesgue Theorem
Now we turn to the Fatou-Lebesgue theorem. Let be a sequence of integrable functions (this time we do not assume they are non-negative) and be some other function which dominates this sequence in absolute value. That is, we have a.e. for all . We define the functions
These two functions are integrable, and we have the sequence of inequalities
Again, this is often stated for a sequence of measurable functions, but the dominated convergence theorem allows us to immediately move to the integrable case. In fact, if the sequence converges pointwise a.e., then a.e. and the inequality collapses and gives us exactly the dominated convergence theorem back again.
Since dominates the sequence , the sequence will be non-negative. Fatou’s lemma then tells us that
Cancelling the integral of we find the first asserted inequality. The second one is true by the definition of limits inferior and superior. The third one is essentially the same as the first, only using the non-negative sequence .