LAUSR

Given a sequence {fn}n∈N$\{f_{n}\}_{n \in \mathbb {N}}$ of measurable functions on a σ-finite measure space such that the integral of each fn$f_{n}$ as well as that of lim supn↑∞fn$\limsup_{n \uparrow\infty} f_{n}$ exists in R‾$\overline{\mathbb {R}}$, we provide a sufficient condition for the following inequality to hold: lim supn↑∞∫fndμ≤∫lim supn↑∞fndμ.$$ \limsup_{n \uparrow\infty} \int f_{n} \,d\mu\leq \int\limsup_{n \uparrow\infty} f_{n} \,d\mu. $$ Our condition is considerably weaker than sufficient conditions known in the literature such as uniform integrability (in the case of a finite measure) and equi-integrability. As an application, we obtain a new result on the existence of an optimal path for deterministic infinite-horizon optimization problems in discrete time.