(a) If (fk) is a bounded sequence in M[a, b] and fk → f a.e., show that f ∈ M[a, b]. [Use the Dominated Convergence Theorem.]
(b) If (gk) is any sequence in M[a, b] and if fk := Arctan ο gk, show that (fk) is a bounded sequence in M[a, b].
(c) If (gk) is a sequence in M[a, b] and if gk → g a.e., show that g ∈ M[a, b].