Proposition 9

— *Let F be a finite-dimensional Banach space. Every measure m on T with values in F is a measure with base $|m|$. If $m = f \cdot |m|$, then $|f(t)| = 1$ locally almost everywhere for $|m|$. For m to have base $\mu$, it is necessary and sufficient that $|m|$ have base $\mu$, and if $m = g \cdot \mu$ then $|m| = |g| \cdot \mu$.

Let $(e_i)_{1 \leq i \leq n}$ and $(e'_i)_{1 \leq i \leq n}$ be dual bases of F and F' (A, II, §2, No. 6) with $|e'_i| = 1$ for all i. Then $|e'_i \circ m| \leq |m|$ for every index i, therefore (Ch. V, §5, No. 5, Th. 2) $e'_i \circ m = h_i \cdot |m|$, where $h_i$ is bounded and $|m|$-measurable. Setting $h = \sum_{i=1}^n h_i \cdot e_i$, we therefore have $m = h \cdot |m|$. If $m = f \cdot |m|$, Prop. 8 shows that $|m| = |f| \cdot |m|$, whence $|f(t)| = 1$ locally almost everywhere for $|m|$ (Ch. V, §5, No. 3, Cor. 2 of Prop. 3). The final assertion follows at once from Prop. 8.