Exercise 12, § 3

Let $E$ be the non-metrizable compact space obtained by equipping the interval $[ -1, +1 ]$ of $\mathbf{R}$ with the topology $\mathscr{T}$ defined in Exer. 13 a) of GT, IX, §2; let $\varphi$ be the mapping $x \mapsto |x|$ of $E$ onto $I = [0, 1]$ (equipped with the topology induced by that of $\mathbf{R}$).

a) Show that $\varphi$ is continuous and that, denoting by $\mathscr{F}$ the set of subsets of $E$ of the form $\varphi^{-1}(A)$, where $A$ is a Borel subset of $I$, the Borel subsets of $E$ are the subsets $B$ such that there exists an $M \in \mathscr{F}$ for which $B \cap \mathbf{C}M$ and $M \cap \mathbf{C}B$ are countable (observe that every open set of $E$ belongs to $\mathscr{F}$).

b) Show that for a numerical function $f$ defined on $E$ to be continuous (for $\mathscr{T}$), it is necessary and sufficient that it be regulated and that $f(x-) = f((-x)+)$ for every $x \in E$. One therefore defines a positive measure $\mu$ on $E$ by setting $\int f\, d\mu = \int_0^1 f(x)\, dx$. Show that the image of $\mu$ under $\varphi$ is the Lebesgue measure on $I$, and the $\mu$-negligible sets are the subsets negligible for Lebesgue measure on $[ -1, +1 ]$.

c) Deduce from a) and b) that there does not exist a $\mu$-measurable section for the $\mu$-measurable equivalence relation $\varphi(x) = \varphi(y)$ in $E$.