Exercise 6, § 4

(a) Let $S$ be a hyperstonian compact space with no isolated point (Integration, Chapter V, §5, Exercise 14). Let $C_0(S)$ be the vector space of realvalued functions, finite or otherwise, defined on S, continuous on S and such that the sets $f(-\infty)$ and $f(+\infty)$ are nowhere dense in S (Integration, Chapter II, § 1, Exercise 13 (g)). Show that, for every measure $\mu > 0$ on S, there exist functions $f$ in $C_0(S)$ whose upper integral is infinite (note that for every nowhere dense closed subset N of S there exists a function $f$ in $C_0(S)$ which is equal to $+\infty$ on N; show then that attention may be restricted to the case where the measure $\mu$ is normal and define $f$ suitably as the upper bound of a sequence of functions in $C_0(S)$).

(b) Let G be the ordered subgroup of $C_0(S)$ consisting of the functions in $C_0(S)$ whose values in $\mathbf{Z}$ or $\pm \infty$. Show that G is a complete lattice and is not isomorphic (as an ordered group) to any subgroup of a product group $\mathbf{R}^l$.

(c) Let K be a field and $A_s = K[[X_s]]_{s \in S}$ the ring of formal power series with coefficients in K in a family of indeterminates $(X_s)$ with the space S as indexing set (Algebra, Chapter IV, § 5, Exercise 1). Let b be the set of formal power series $P \in A_s$ with the following property: there exists a nowhere dense set $N \subset S$ such that every monomial of P whose coefficient is $\neq 0$ contains an $X_s$ for which $s \in N$. Show that b is an ideal of $A_0$ and that the ring $A = A_0/b$ is an integral domain and completely integrally closed (to prove the latter point, apply Chapter V, § 1, no. 4, Proposition 14, arguing as in Chapter V, § 1, Exercise 10; use also the fact that in S every meagre set is nowhere dense).

(d) Let $f$ be an element $\geq 0$ of the group G and let N be the nowhere dense set of points where $f(s) = \pm \infty$. Let $p_f$ be the element of A the image of the formal power series $\prod_{s \in N} (1 + X_s)^{f(s)}$; the mapping $f \mapsto p_f$ is an isomorphism of the additive monoid $G_+$ onto a multiplicative submonoid of A. Deduce that A cannot be the intersection of a family of valuation rings of height 1 of its field of fractions (show that this would imply that G is isomorphic to a subgroup of a product $\mathbf{R}^l$). (Cf. Exercises 8 and 9.)