Exercise 17, § 5

Let $X$ be a compact space, $\mu$ a positive measure on $X$. A set $\mathscr{P}$ of finite partitions of $X$ into integrable sets, directed for the relation «$\varpi$ is coarser than $\varpi'$ », is said to be fundamental if, for every entourage $V$ of the uniform structure of $X$, there exists a partition $\varpi = (A_i)$ of $X$ belonging to $\mathscr{P}$ such that all of the $A_i$ are small of order $V$. For every finite partition $\varpi = (A_k)$ belonging to $\mathscr{P}$, and every bounded numerical function $f$ on $X$, set

(the 'Riemann sums' relative to $f$ and the partition $\varpi$).

a) Show that $s_{\varpi}(f) \leq \mu_*(f) \leq \mu^*(f) \leq S_{\varpi}(f)$ for every partition $\varpi \in \mathscr{P}$, and that $s_{\varpi}(f)$ and $S_{\varpi}(f)$ each tend to a limit with respect to the directed ordered set $\mathscr{P}$.

b) If $\mathscr{P}$ is the (fundamental) set of all finite partitions of $X$ into integrable sets, show that for every function $f$ that is bounded and integrable, $s_{\varpi}(f)$ and $S_{\varpi}(f)$ tend to $\int f d\mu$ with respect to $\mathscr{P}$.

c) If $f$ is a bounded function that is continuous almost everywhere in $X$, then $s_{\varpi}(f)$ and $S_{\varpi}(f)$ tend to $\int f d\mu$ with respect to every fundamental set $\mathscr{P}$ of finite partitions of $X$ into integrable sets (for every $\varepsilon > 0$, consider the closed set $A$ of points where the oscillation of $f$ is $\geq \varepsilon$, and for every partition $\varpi \in \mathscr{P}$ whose sets are small of order $V$, consider separately those sets of $\varpi$ that intersect $V(A)$ and those that do not intersect it). If $f$ is bounded and lower semi-continuous on $X$, show that $s_{\varpi}(f)$ tends to $\int f d\mu$ with respect to every fundamental set $\mathscr{P}$ of finite partitions of $X$ into integrable sets (regard $f$ as the upper envelope of continuous functions).

d) A set $A \subset X$ is said to be quadrable (for $\mu$) if its characteristic function is continuous almost everywhere, or, what amounts to the same, if its boundary is $\mu$-negligible. Show that every point $x_0$ of $X$ has a fundamental system of quadrable open neighborhoods (for every neighborhood $V$ of $x_0$, let $f$ be a continuous function with values in $[0,1]$, equal to 1 at the point $x_0$ and to 0 on $\mathbf{C}V$; consider the sets of the $x$ such that $f(x) > \alpha$, for $0 < \alpha < 1$). From this, deduce that there exists a fundamental set $\mathscr{P}$ of finite partitions of $X$, such that every partition $\varpi \in \mathscr{P}$ consists of open sets and negligible sets.

e) Let $\mathscr{P}$ be a fundamental set of finite partitions of $X$ into integrable sets, such that every partition $\varpi \in \mathscr{P}$ consists of open sets and negligible sets. For every bounded function $f$ on $X$, let $g$ be the largest lower semi-continuous function on $X$ that is $\leq f$ (GT, IV, §6, Prop. 4); show that for every partition $\varpi \in \mathscr{P}$, $s_{\varpi}(f) = s_{\varpi}(g)$. From this, deduce that for $s_{\varpi}(f)$ and $S_{\varpi}(f)$ to tend to a common limit with respect to $\mathscr{P}$, it is necessary that $f$ be continuous almost everywhere in $X$.

$f$ Deduce from e) an example of a negligible function $f$ and a fundamental set $\mathscr{P}$ of finite partitions of $X$ into integrable sets, such that $s_{\varpi}(f)$ and $S_{\varpi}(f)$ do not tend to the same limit with respect to $\mathscr{P}$ (take for $X$ the interval $[0,1]$ of $\mathbf{R}$, and for $\mu$ the Lebesgue measure).