8. Show that a set function on a $\sigma$-algebra is a measure if and only if it is a premeasure.
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A set function $\mu$ on a $\sigma$-algebra $\mathcal{M}$ is a **measure** if it satisfies: 1. $\mu(A) \ge 0$ for all $A \in \mathcal{M}$. 2. $\mu(\emptyset) = 0$. 3. Countable additivity: For any countable collection of disjoint sets $\{A_i\}_{i=1}^\infty$ in Show more…
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