  It is well-known that every weakly convergent sequence in $\ell_1$ is
convergent in the norm topology (Schur's lemma). Phillips' lemma asserts even
more strongly that if a sequence $(\mu_n)_{n\in\mathbb N}$ in $\ell_\infty'$
converges pointwise on $\{0,1\}^\mathbb N$ to $0$, then its $\ell_1$-projection
converges in norm to $0$. In this note we show how the second category version
of Schur's lemma, for which a short proof is included, can be used to replace
in Phillips' lemma $\{0,1\}^\mathbb N$ by any of its subsets which contains all
finite sets and having some kind of interpolation property for finite sets.
