Interchanging a limit and an integral: necessary and sufficient conditions

Let {f_n}_n ∈ℕ be a sequence of integrable functions on a σ -finite measure space (Ω, ℱ, μ ) . Suppose that the pointwise limit lim_n ↑∞ f_n exists μ -a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: lim_n ↑∞∫ f_n dμ = ∫lim_n ↑∞ f_n dμ.