Exercise 17, § 1

Let $T$ be a locally compact space, $S$ a subset of $T$ equipped with a law of composition $(x, y) \mapsto xy$ that makes it a monoid$^1$ (not necessarily having a priori a neutral element), and which is continuous on $S \times S$ when $S$ is equipped with the topology

$^1$ Monoïde, as defined in the early editions of Alg., Ch. I, is not required to have a neutral element, as it is in the bound edition of Algèbre; what is meant here is an associative magma (A, I, §1, No. 3, Def. 5).

induced by that of T. Assume in addition that every element of S is cancellable² (A, I, §2, No. 2). Let $\mu$ be a bounded positive measure on T, concentrated on S (so that S is $\mu$-measurable) and of total mass 1. Assume that $\mu$ is left invariant in the following sense: for every numerical function $f$ defined on S, continuous and bounded (hence $\mu$-measurable by Ch. IV, §5, No. 5, Cor. of Prop. 8), one has $\int f(sx)\, d\mu(x) = \int f(x)d\mu(x)$ for all $s \in S$. It comes to the same to say that $\mu(sK) = \mu(K)$ for every compact subset K of S.

a) Consider the product measure $\mu \otimes \mu$ on $T \times T$; show that there exist compact subsets K of S such that the images of $K \times K$ under the two continuous mappings $(x, y) \mapsto (x, xy)$ and $(x, y) \mapsto (xy, x)$ have measure arbitrarily close to 1 (use the Lebesgue–Fubini theorem). Conclude from this that these images have a common point, and deduce therefrom that S has a neutral element (cf. A, I, §2, Exer. 9).

b) Show that S is a compact group. (First prove that for every $x \in S$, the inner measure $\mu_*(xS)$ is equal to 1, hence that $xS$ is measurable and that the measure is concentrated on $xS$; $xS$ is then a monoid to which the result of a) can be applied, which shows that S is a group. Argue as in Prop. 2 to prove that S is compact. Finally, make use of Exer. 21 of GT, III, §4.³)