Fatou's Lemma in Several Dimensions

Abstract

In this note the following generalization of Fatou's lemma is proved: Lemma. Let {fn)n_l be a sequence of integrable functions on a mea- sure space S with values in R+, the nonnegative orthant of a d-dimen- sional Euclidean space, for which ffn—*aGiR+. Then there exists an in- tegrable function f, from S to R+, such that a.e. f(s) is a limit point of VnisV^and ff^a. 1. Introduction. When d = l, the result is a form of Fatou's lemma. The assertion above is applied in mathematical economics (4).2 It is also strongly connected with the theory of set valued functions (2) or correspondences (3). The nontrivial part of the arguments is lim- ited to the case where 5 is an atomless measure space. In the purely atomic case the lemma is reduced to a simple exercise in series. In any case, the lemma cannot be proved by a successive application of Fatou's lemma d times. A few corollaries of the lemma are proved in §3. 2. Preliminary results and the proof of the lemma. Let (A)°=1 be a sequence of (nonempty) subsets of Rd. We denote by Lim Sup^4 the set of all the limit points of the sequences (a),T-i with anEAn, » = 1, 2, • • ■ . Denote by x-y the inner product, Yt=i xlyl, in Rd.