Lemma 2

Every separable subspace $G$ of the Banach space $L^\infty(T, \mu)$ has the lifting property.

By the hypothesis, there exists a countable dense subset $H$ of $G$ that is a linear subspace with respect to the field $\mathbf{Q}$ of rational numbers; let $(h_n)$ be a (countable) basis of $H$ over $\mathbf{Q}$. For every integer $n$, let $h'_n$ be an element of $\mathscr{L}^\infty$ such that $\pi(h'_n) = h_n$, and let $\rho'$ be the $\mathbf{Q}$-linear mapping of $H$ into $\mathscr{L}^\infty$ defined by $\rho'(h_n) = h'_n$; it is clear that $\pi \circ \rho'$ is the identity on $H$. Moreover, for every $h \in H$ one has $|\rho'(h)(t)| \leq N_\infty(h)$ except at the points $t$ of a locally negligible set $A(h)$. Let $A$ be the union of the $A(h)$ for $h \in H$, which is also locally negligible. For every $h \in H$, denote by $\rho(h)$ the function $h'' \in \mathscr{L}^\infty$ such that $h''(t) = \rho'(h)(t)$ if $t \notin A$, and $h''(t) = 0$ if $t \in A$. It is clear that $\rho$ is a $\mathbf{Q}$-linear mapping of $H$ into the subspace $\mathscr{B}$ of bounded functions in $\mathscr{L}^\infty$, such that $\pi \circ \rho$ is the identity on $H$ and such that $|\rho(h)(t)| \leq N_\infty(h)$ for all $h \in H$ and $t \in T$. Since $\mathscr{B}$ is a Banach space for the norm $\|f\| = \sup_{t \in T} |f(t)|$ (Ch. IV, §5, No. 4, Th. 2), $\rho$ may be extended to a continuous $\mathbf{R}$-linear mapping, again denoted $\rho$, of $G$ into $\mathscr{B}$, which is obviously a lifting of $G$.