Example

— Let $\mu$ be Lebesgue measure on $\mathbf{R}$, I any interval in $\mathbf{R}$; I is a locally compact subspace of $\mathbf{R}$, and the measure induced by $\mu$ on I is the linear form

on $\mathscr{K}(I; \mathbf{C})$, where $a$ and $b$ are the endpoints (finite or not) of I (cf. § 4, No. 4, Example), in other words, what we have called Lebesgue measure on I.

When Y is an open subspace of X, Def. 4 coincides with the definition of the measure induced by $\mu$ on Y (or the restriction of $\mu$ to Y) given in Ch. III, §2, No. 1: indeed, for every function $g \in \mathscr{K}(Y; \mathbf{C})$ the function $g'$ is then continuous on X.

We shall study integration with respect to an induced measure in detail in Ch. V, §7, and until then we shall need only the following results: