Remark 2

Suppose that I is countably additive; let $(A_n)_{n \in \mathbf{N}}$ be an increasing sequence of Borel sets, and let $A = \bigcup_{n \in \mathbf{N}} A_n$. The sets $D_0 = A_0$, $D_n = A_n - A_{n-1}$ being pairwise disjoint, and their union being A, we have $I(A) = \sum_n I(D_n) = \lim_{n \to \infty} I(A_n)$. Similarly, if $(B_n)$ is a decreasing sequence of Borel sets, and if $I(B_0) < +\infty$, then $I(\bigcap_n B_n) = \lim_{n \to \infty} I(B_n)$: it suffices to apply the preceding to the sets $A_n = B_0 - B_n$.