Exercise Let and in measure. Show that,
- (a) .
- (b) in .
- (a) In Exercise 33 we proved a theorem analogous to Fatou’s lemma for in measure. We will use this analogue of Fatou’s lemma to prove an analogue of dominated convergence theorem. By taking real and imaginary parts, we can assume that and are real-valued. Applying Exercise 33 to yields,
And applying Exercise 33 to yields,
Therefore we have . Note that this almost exactly the same as the normal proof of dominated convergence theorem from Fatou’s lemma.
- (b) It is easy to see that we have in measure and a.e. Therefore we apply (a) to get that so in .