Definition 1

One calls integral of $f$ with respect to $\mu$, and denotes by $\int f d\mu$ or $\int f(t) d\mu(t)$, the element of ${F'}^*$ defined by

for all $z' \in F'$.

If $f$ is continuous with compact support, it is scalarly integrable and Def. 1 coincides with the definition of the integral of $f$ given in Ch. III, §3, No. 1. On the other hand, if $F$ is a Banach space and $f$ is essentially integrable (Ch. V, §1, No. 3, Def. 3), then $f$ is scalarly essentially integrable and Def. 1 coincides with the definition of the integral of $f$ given in Chapter V (Ch. V, §1, No. 3 and Ch. IV, §4, No. 2, Cor. 1 of Th. 1).