. Let \((X, \mathscr{A}, \mu)\) be a finite measure space and \(\left(A_{n}ight)_{n \in \mathbb{N}},\left(B_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) such that \(A_{n} \supset B_{n}\) for all \(n \in \mathbb{N}\). Show that

\[\mu\left(\bigcup_{n \in \mathbb{N}} A_{n}ight)-\mu\left(\bigcup_{n \in \mathbb{N}} B_{n}ight) \leqslant \sum_{n \in \mathbb{N}}\left(\mu\left(A_{n}ight)-\mu\left(B_{n}ight)ight)\]

[show first that \(\bigcup_{n} A_{n} \backslash \bigcup_{k} B_{k} \subset \bigcup_{n}\left(A_{n} \backslash B_{n}ight)\) then use Proposition 4.3 (viii).]

Data from proposition 4.3

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Let (X, A,p) be a measure space and A, B, A, BA, nN. Then (i) An B=0 (AUB) = (A) +(B) (ii) ACB (A) p(B) (iii) ACB, (A)