For a subset $S \subseteq X$, let $\complement \left({S}\right)$ denote the relative complement of $S$ in $X$. We first prove the second property of an algebra of sets, as described on that page. Let $S$ be $\mu^*$-measurable. For any subset $A \subseteq X$: {{begin-eqn}} {{eqn | l = \mu^* \left({A}\right)       | r = \mu^* \left({A \cap S}\right) + \mu^* \left({A \cap \complement \left({S}\right)}\right) }} {{eqn | r = \mu^* \left({A \cap \complement \left({\complement \left({S}\right)}\right)}\right) + \mu^* \left({A \cap \complement \left({S}\right)}\right)       | c = Complement of Complement }} {{end-eqn}} as desired. Now we prove the first property. Suppose that $S_1$ and $S_2$ are $\mu^*$-measurable sets.  Let $A$ be any subset of $X$.   Since: {{begin-eqn}} {{eqn | l = A \cap \left({S_1 \cup S_2}\right)       | r = \left[{\left({A \cap \left({S_1 \cup S_2}\right)}\right) \cap S_1}\right] \cup \left[{\left({A \cap \left({S_1 \cup S_2}\right)}\right) \setminus S_1}\right]       | c = Set Difference Union Intersection and Union is Commutative }} {{eqn | r = \left[{A \cap \left({\left({S_1 \cup S_2}\right) \cap S_1}\right)}\right] \cup \left[{\left({\left({S_1 \cup S_2}\right) \cap A}\right) \setminus S_1}\right]       | c = Intersection is Associative and Intersection is Commutative }} {{eqn | r = \left[{A \cap \left({\left({S_1 \cup S_2}\right) \cap S_1}\right)}\right] \cup \left[{\left({\left({S_1 \cup S_2}\right) \setminus S_1}\right) \cap A}\right]       | c = Intersection with Set Difference is Set Difference with Intersection }} {{eqn | r = \left({A \cap S_1}\right) \cup \left({\left({S_2 \setminus S_1}\right) \cap A}\right)       | c = Intersection Absorbs Union and Set Difference with Union is Set Difference }} {{eqn | r = \left({A \cap S_1}\right) \cup \left({\left({S_2 \cap A}\right) \setminus S_1}\right)       | c = Intersection with Set Difference is Set Difference with Intersection }} {{eqn | r = \left({A \cap S_1}\right) \cup \left({\left({A \cap S_2}\right) \setminus S_1}\right)       | c = Intersection is Commutative }} {{eqn | r = \left({A \cap S_1}\right) \cup \left({\left({A \setminus S_1}\right) \cap S_2}\right)       | c = Intersection with Set Difference is Set Difference with Intersection }} {{end-eqn}} {{wtd|I'm pretty sure the above, or at least significant parts of the above, have/has already been posted up as proofs in their own right. If I remember, or rather, when I'm in the mood, I may go through and find them.}} we have, by the subadditivity of an outer measure: :$\mu^* \left({A \cap \left({S_1 \cup S_2}\right)}\right) \leq \mu^* \left({A \cap S_1}\right) + \mu^* \left({\left({A \setminus S_1}\right) \cap S_2}\right)$ Thus: {{begin-eqn}} {{eqn | r = \mu^* \left({A \cap \left({S_1 \cup S_2}\right)}\right) + \mu^* \left({A \setminus \left({S_1 \cup S_2}\right)}\right) }} {{eqn | r = \mu^* \left({A \cap \left({S_1 \cup S_2}\right)}\right) + \mu^* \left({\left({A \setminus S_1}\right) \setminus S_2}\right)       | c = Set Difference with Union }} {{eqn | o = \le       | r = \mu^* \left({A \cap S_1}\right) + \mu^* \left({\left({A \setminus S_1}\right) \cap S_2}\right) + \mu^* \left({\left({A \setminus S_1}\right) \setminus S_2}\right)       | c = by the above argument }} {{eqn | r = \mu^* \left({A \cap S_1}\right) + \mu^* \left({A \setminus S_1}\right)       | c = Definition of Measurability of $S_2$ }} {{eqn | r = \mu^* \left({A}\right)       | c = Definition of Measurability of $S_1$ }} {{end-eqn}} The result follows by the subadditivity of an outer measure. {{refactor}} Alternatively, one could use the equality {{begin-eqn}} {{eqn | l = \mu^* \left({A \cap \left({S_1 \cup S_2}\right)}\right)       | r = \mu^* \left({A \cap \left({S_1 \cup S_2}\right) \cap S_1}\right) + \mu^* \left({A \cap \left({S_1 \cup S_2}\right) \setminus S_1}\right) }} {{eqn | r = \mu^* \left({A \cap S_1}\right) + \mu^* \left({A \cap S_2 \setminus S_1}\right) }} {{end-eqn}} to prove the result directly without the use of subadditivity. {{qed}} Category:Measure Theory Category:Set Systems qv27l4hn8xkf63xmse0u1ewxu2nzud2