Proposition 7 ('associativity of the product of measures')

— Let $(I_k)_{1 \leq k \leq r}$ be a partition of the interval $[1, n]$ of $\mathbf{N}$; then

For, these two measures coincide, by (12), for every function in $\bigotimes_{i=1}^n \mathscr{K}(X_i; \mathbf{C})$.

The integral of a function $f \in \mathscr{K}(X; \mathbf{C})$ with respect to the product measure is denoted

or or or also or with $n$ signs $\int$; it is said to be a multiple integral of order $n$, or an $n$-tuple integral. By virtue of the associativity of the product of measures and the theorem on inverting the order of integration (No. 1, Th. 2), we have, for every permutation $\sigma$ of $[1, n]$, (14)

The integral notation and formula (14) may be extended in an obvious way to functions $f \in \mathscr{K}(X; E)$ with values in a Hausdorff locally convex space $E$, such that $f(X)$ is contained in a complete convex subset of $E$. We leave to the reader the task of generalizing to the product of any finite number of measures the results of Nos. 2 and 3 concerning the product of two measures.

In particular, one calls Lebesgue measure on $\mathbf{R}^n$ the product of $n$ measures identical to the Lebesgue measure on $\mathbf{R}$; the integral of a function $f \in \mathscr{K}(\mathbf{R}^n; E)$, satisfying the preceding condition, is denoted

and is equal to Lebesgue measure on $\mathbf{R}^n$ is invariant under every translation.