Exercise 18, § 2

a) Let $T$ be a locally compact space, $(K_\alpha)$ a locally finite covering of $T$ by compact sets. Let $\mu$ be a positive measure on $T$, $\mu_\alpha$ the measure induced by $\mu$ on $K_\alpha$. Assume that each of the spaces $L^\infty(K_\alpha, \mu_\alpha)$ has the lifting property. Show that $L^\infty(T, \mu)$ has the lifting property.

b) Let $K$ be a metrizable compact space, $\mu$ a positive measure on $K$, $(\varpi_n)$ a fundamental sequence (Ch. IV, §5, Exer. 17) of finite partitions of $K$ into integrable sets. For every integrable subset $A$ of $K$ and every element $\tilde{f} \in L^1(\mu)$, set $\lambda_\alpha(\tilde{f}) = 0$ if $\mu(A) = 0$, and $\lambda_\alpha(\tilde{f}) = (\mu(A))^{-1} \int_A f d\mu$ otherwise; for every $n$, set $\rho_n(\tilde{f}) = \sum_k \lambda_{A_k}(\tilde{f}) \varphi_{A_k}$, if $\varpi_n = (A_k)$. Show that the sequence $(\rho_n(\tilde{f}))$ of $\mu$-integrable functions converges almost everywhere to $f$. (Consider first the case that $f$ is a continuous function. Given an arbitrary number $a > 0$, show next that the union $B$ of all the sets $A$ belonging to at least one of the partitions $\varpi_n$ and such that $a \cdot \mu(A) \leq \int_A f d\mu$, is measurable and is such that $\mu(B) \leq a^{-1} \int f d\mu$. Then approximate $f$ in $L^1$ by a sequence of continuous functions.)

c) Show that every metrizable compact space has the lifting property. (Let $\mathfrak{U}$ be an ultrafilter on $\mathbf{N}$ finer than the Fréchet filter; show that $\rho(\tilde{f}) = \lim_{\mathfrak{U}} \rho_n(\tilde{f})$ is a lifting of $L^\infty$.)

d) Deduce from a) and c) that every metrizable locally compact space has (for any positive measure whatever) the lifting property.