Example 1

Suppose that T is a discrete space, and that μ is the measure on T defined by placing a mass +1 at each point of T (Ch. III, §1, No. 3). Let h be a function $\geqslant 0$ defined on T; since the function h is lower semi-continuous (even continuous) on T, $\mu^*(h) = \mu^\bullet(h) = \sum_{t \in T} h(t)$ (Ch. IV, §1, No. 1, Example).

For the measure μ, the notions of integrable function and essentially integrable function are therefore identical. This being so, to say that a mapping $t \mapsto \lambda_t$ of T into $\mathscr{M}_+(X)$ is scalarly essentially μ-integrable amounts to saying that the family $(\lambda_t)_{t \in T}$ is summable (§2, No. 1), and one then has $\int \lambda_t\, d\mu(t) = \sum_{t \in T} \lambda_t$. Note that the mapping $t \mapsto \lambda_t$ is vaguely continuous.