Proposition 9

Let $F$, $F'$ and $G$ be three Banach spaces, and let $(x, y) \mapsto [x \cdot y]$ be a continuous bilinear mapping of $F \times F'$ into $G$. Let $f$ (resp. $f'$) be a function defined on $T$ (resp. $T'$) with values in $F$ (resp. $F'$) and essentially integrable for $\mu$ (resp. $\mu'$). Let $g$ be the function $(t, t') \mapsto [f(t) \cdot f'(t')]$; then $g$ is essentially integrable for $\mu \otimes \mu'$, and

If, moreover, $f$ and $f'$ are integrable, then $g$ is integrable.

The function $(t, t') \mapsto [\mathbf{f}(t) \cdot \mathbf{f}'(t')]$ is $(\mu \otimes \mu')$-measurable by Cor. 1 of Prop. 3. On the other hand, if $b$ denotes the norm of the bilinear mapping $(\mathbf{x}, \mathbf{y}) \mapsto [\mathbf{x} \cdot \mathbf{y}]$, then

by virtue of Prop. 8. This shows that $[\mathbf{f}(t) \cdot \mathbf{f}'(t')]$ is essentially integrable for $\mu \otimes \mu'$ (§ 1, No. 3, Prop. 9). Suppose that $\mathbf{f}$ and $\mathbf{f}'$ are integrable: then $\mathbf{f}$ and $\mathbf{f}'$ are moderated, and $\mathbf{g}$ is moderated (Cor. 2 of Prop. 5) therefore integrable (§ 1, No. 3, Cor. of Prop. 9). In this case the formula (14) follows from the Lebesgue–Fubini theorem and the linearity of the integral (Ch. IV, § 4, No. 2, Th. 1). To complete the treatment of the case that $\mathbf{f}$ and $\mathbf{f}'$ are essentially integrable, one then applies (14) to two integrable functions $\mathbf{f}_1$ and $\mathbf{f}_1'$, equal locally almost everywhere to $\mathbf{f}$ and $\mathbf{f}'$, on observing that $[\mathbf{f} \cdot \mathbf{f}'] = [\mathbf{f}_1 \cdot \mathbf{f}_1']$ locally almost everywhere in $T \times T'$ (Prop. 4).

This result extends to the product of complex measures.