Remark 1

The relation $(\mu_X)^\bullet = (\mu^\bullet)_X$ permits using the notation $\mu_X^\bullet$ without ambiguity; we shall do so henceforth. The preceding Prop. 1 and Prop. 2 of §1, No. 2 show that the measures denoted $\mu_K$ until now, for $K$ compact, are indeed induced measures in the sense of Def. 1. Similarly, if $T$ is locally compact, and if $X$ is a locally compact subspace of $T$, the above Prop. 1 and Prop. 1 of Ch. V, §7, No. 1 show that Def. 1 coincides with the definition of Ch. IV, §5, No. 7.