Generalized Egoroff’s theorem

This note is closely related to the paper [R. Pinciroli: On theindependence of a generalized statement of Egoroff’s theorem from ZFC afterT. Weiss, Real Anal. Exchange 32 (2006-2007), 225-232] and it presents slight improvements of its results. Theorem 1.13 shows a connection with Galois-Tukey embeddings; Corollary 1.14 presents another inequality which is dual to the previously known one; Corollary 3.5 shows that there is no distinction between positive outer measure and full outer measure in the given context; and Corollary 4.3 unifies the known counterexamples.