Proposition 2

— Let f be a scalarly essentially $\mu$-integrable mapping of T into F. For every numerical function $g \geqslant 0$ that is $\mu$-measurable and bounded, the mapping $t \mapsto g(t)f(t)$ (denoted gf or fg) of T into F is scalarly essentially $\mu$-integrable, f is scalarly essentially $(g \cdot \mu)$-integrable, and

This is an immediate consequence of the formula $\langle \mathbf{z}', gf \rangle = g \langle \mathbf{z}', f \rangle$, for all $\mathbf{z}' \in F'$, and the formula $\int h d(g \cdot \mu) = \int hg d\mu$ for every essentially $\mu$-integrable scalar function $h$ (Ch. V, §5, No. 3, Th. 1).

A great many propositions about essentially integrable numerical functions may be transposed word for word into propositions about scalarly essentially integrable vector-valued functions. Among the more important, we call attention to the conditions for a function to be essentially integrable with respect to a measure defined by a density (Ch. V, §5, No. 3, Th. 1), or with respect to the image of a measure (Ch. V, §6, No. 2, Th. 1), or with respect to an induced measure (Ch. V, §7, No. 1, Th. 1), or with respect to the sum of a summable family of positive measures (Ch. V, §2, No. 2, Props. 1 and 3 and Cor. 3 of Prop. 1). These transpositions are left to the reader.

However, to obtain statements corresponding to the theorems on 'double integrals' (Ch. V, §3, No. 3, Th. 1 and §8, No. 4, Th. 1 (Lebesgue–Fubini theorem)), it is necessary to strengthen the hypotheses (cf. Exer. 1); on applying the previously cited theorems to each of the functions $\langle \mathbf{z}', f \rangle$, where $\mathbf{z}' \in F'$, one thus obtains the following propositions: