Discussion
Let \(\mathbf{F}\) be a \(\sigma\)-field of subsets of \(\Omega\). Let \(\mu\) be a non-negative, finitely additive set function on \(\mathbf{F}\).
Suppose \(\{A_{n}\}\) is a sequence of disjoint sets such that \(\bigcup_{n=1}^{\infty} A_{n} \in \mathbf{F}\).
Then which of the following statements is always correct ?
Answer: D
The statement always correct is \(\sum \mu(A_{n}) \le \mu\left(\bigcup A_{n}\right)\).
Let \(A = \bigcup A_{n}\). For each \(N\), finite additivity and monotonicity (which follows from finite additivity and non-negativity) give
\[ \begin{aligned} \sum_{n=1}^{N} \mu(A_{n}) &= \mu\left(\bigcup_{n=1}^{N} A_{n}\right) \\ &\le \mu(A) \end{aligned} \]
Letting \(N \to \infty\) gives the inequality. Equality (option A) is countable additivity, which a merely finitely additive function need not have; so the reverse inequality B can also fail. Option C is wrong because the complements \(A_{n}^{c}\) are not disjoint.
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