I'm reading Probability: Theory and Examples by Rick Durrett. Theorem 1.1.1 states that
Let $\mu$ be a measure on $(\Omega, \mathcal F)$
(i) monotonicity. If $A \subset B$ then $\mu(A) \le \mu(B)$.
(ii) subadditivity. If $A \subset \cap_{m=1}^\infty A_m$, then $\mu(A) \le \sum_{m=1}^\infty \mu(A_m)$
Proof. (i) Let $B − A = B \cap A^c$ be the difference of the two sets ...
The proof of (ii) given in the book is like this
Let A_n' = A_n \cap A, B_1 = A_1' and for $n > 1$, B_n = A_n' − \cap_{m=1}^{n−1} (A_m')^c . Since the $B_n$ are disjoint and have union $A$ we have using (i) of the definition of measure, $B_m \subset A_m$ , and (i) of this theorem $\mu(A) = \sum_{m=1}^\infty \mu(B_m) \le \sum_{m=1}^\infty \mu (A_m)$
I can't understand why
the $B_n$ are disjoint and have union $A$
Is there a typo in the definition of $B_n$?