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Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211–256) states that if X is separable, then the Borel class of T−1 is α if, and only if, X* is the αth iterated sequential weak*-closure of T*Y* for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683–699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.
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