Fatou’s lemma in several dimensions

Abstract:

In this note the following generalization of Fatou’s lemma is proved: Lemma. Let $({f_n})_{n - 1}^\infty$ be a sequence of integrable functions on a measure space $S$ with values in $R_ + ^d$, the nonnegative orthant of a $d$-dimensional Euclidean space, for which $\int {{f_n} \to a \in R_ + ^d}$. Then there exists an integrable function $f$, from $S$ to $R_ + ^d$, such that a.e. $f(s)$ is a limit point of $({f_n}(s))_{n - 1}^\infty$ and $\int {f \leqq a}$.