Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f n ’s is involved and we assume f n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f n ) to f is assumed.