Proposition 10

Let $\nu$ be the product premeasure of $\lambda$ and $\mu$; for every function $f \in \mathscr{F}_+(S)$ and every function $g \in \mathscr{F}_+(T)$,

The premeasure $\nu$ is the only premeasure on $S \times T$ that satisfies (4).

As $K$ (resp. $L$) runs over the set of compact subsets of $S$ (resp. of $T$), we have

by Prop. 8 of Ch. V, §8, No. 3.

Let $\eta$ be a second premeasure on $S \times T$ satisfying (4), and let $K$ and $L$ be compact subsets of $S$ and $T$ respectively, $f$ and $g$ elements of $\mathscr{F}_+(K)$ and $\mathscr{F}_+(L)$ respectively. One has the relation $(f \otimes g)^0 = f^0 \otimes g^0$ between the extensions by 0, therefore (§1, No. 2, Prop. 2)

In particular, if one takes $f \in \mathscr{K}_+(K)$, $g \in \mathscr{K}_+(L)$, one sees that $\eta_{K \times L}$ has the characteristic property of the product measure $\lambda_K \otimes \mu_L$ (Ch. III, §4, No. 1, Th. 1). Therefore $\eta_{K \times L} = \nu_{K \times L}$; since every compact subset of $S \times T$ is contained in a set of the form $K \times L$, the transitivity of induced measures implies that $\eta = \nu$.