Lemma 8

Let U be an open set in $\mathbf{C}^n$. Let $\omega_1$ be a continuous differential form of degree $n$ in U with compact support and with values in A (resp. $\omega_2$ a continuous differential form of degree 2 in $\mathbf{C}$ with compact support and with values in $\mathbf{C}$). Let us denote by $\pi_1$ and $\pi_2$ the canonical projections of $U\times \mathbf{C}$ onto U and $\mathbf{C}$. The differential form $\pi_1^*\omega_1\wedge \pi_2^*\omega_2$ on $U\times \mathbf{C}$ is continuous with compact support and with values in A. We have

Let $\mu_n$ be the Lebesgue measure on $\mathbf{C}^n$ and $\mu_1$ the Lebesgue measure on $\mathbf{C}$. There exist $\psi_1\in \mathscr{K}(U; A)$ and $\psi_2\in \mathscr{K}(\mathbf{C})$ such that the vector measure associated with $\omega_1$ is equal to $\psi_1\cdot \mu_n$, and the vector measure associated with $\omega_2$ is equal to $\psi_2\cdot \mu_1$. The vector measure associated with the differential form $\pi^*_1\omega_1\wedge \pi_2^*\omega_2$ is $(\psi_1\otimes \psi_2)\cdot \mu_n\otimes \mu_1$.

Let $\ell$ be a continuous linear form on A. After INT, VI, §2, n$^o2$, definition 2 and the definition of the product measure (INT, III, §4, n$^o1$, definition 1), it follows that

whence the result (INT VI, loc. cit.).