Exercise 8, § 6

Let T be a compact space, $\mu$ a diffuse positive measure on T of total mass 1.

a) Show that there exists a continuous mapping $\pi$ of T onto E = $[0,1]$ such that the image of $\mu$ under $\pi$ is the Lebesgue measure $\lambda$ on E. (Proceed as in the proof of Urysohn’s theorem (GT, IX, §4, Th. 1) by defining, for every $t$ such that $0 \leq t \leq 1$, an open set $U(t) \subset T$, quadrable for $\mu$ (Ch. IV, §5, Exer. 17), such that $U(0) = \varnothing$, $U(1) = T$, $\overline{U(t)} \subset U(t')$ for $t < t'$, and finally $\mu(U(t)) = t$; one uses Exer. 7 of §5 to prove that if $V, W$ are two quadrable open sets in T such that $\overline{V} \subset W$ and $\mu(V) < \mu(W)$, then there exists a quadrable open set U such that $\overline{V} \subset U \subset \overline{U} \subset W$ and such that

Finally, one applies Exer. 5 a).)

b) Deduce from a) that there exist subsets of T that are not $\mu$-measurable (Ch. IV, §4, Exer. 8), that there exist sequences in $\mathscr{L}^p(T, \mu)$ that convergence in mean of order $p$ to 0 but do not converge to 0 at any point of T, for $1 \leq p < +\infty$ (Ch. IV, §3, Exer. 1), and sequences $(f_n)$ such that $(\tilde{f}_n)$ converges to 0 for the weakened topology of $L^p(T, \mu)$ but does not converge in measure to 0 (§5, Exer. 19).

c) Suppose, moreover, that T is metrizable. Show that there exist a $\mu$-negligible subset N of T, a $\lambda$-negligible subset M of E, and a homeomorphism $\pi$ of E — M onto T — N such that, if one extends (arbitrarily) $\pi$ to a mapping $\varphi$ of E into T, and $\pi^{-1}$ to a mapping $\psi$ of T into E, then $\varphi(\lambda) = \mu$ and $\psi(\mu) = \lambda$. (Using Exer. 17 of Ch. IV, §5, show that for every integer $n > 0$, there exists a finite partition of T formed by a $\mu$-negligible set and by quadrable open sets, of diameter $\leq 1/n$ (for a metric compatible with the topology of T) and of measure $\leq 1/n$. Proceeding recursively, deduce therefrom, by passage to the limit, the existence of a continuous mapping $f$ of E — D into T, where D is a countable subset of E, such that $f(\lambda) = \mu$; show that one can arrange that $f$ be a homeomorphism of E — D onto a subset of T of measure 1 (cf. §8. Exer. 14).)