Interchanging a limit and an integral: necessary and sufficient conditions

Abstract

Abstract Let { f n } n ∈ N $\{f_{n}\}_{n \in \mathbb {N}}$ be a sequence of integrable functions on a σ-finite measure space ( Ω , F , μ ) $(\Omega, \mathscr {F}, \mu )$ . Suppose that the pointwise limit lim n ↑ ∞ f n $\lim_{n \uparrow \infty } f_{n}$ 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 d μ = ∫ lim n ↑ ∞ f n d μ . $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$

Keywords

• σ-finite measure space
• Integrable functions
• Vitali convergence theorem
• Uniform integrability