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We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence $\left(f_n\right)$ 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 $f_n\to f$ almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of $\left(f_n\right)$ to $f$ is assumed.