※ 引述《icebergvodka (肥嘟嘟左衛門)》之銘言：
: 1.
: If u is an outer measure and if every open set is measuable,
: then u is a metric outer measure.
: 3.
: Given an example of a signd measure for which the Hahn decomposition is not
: unique.
與 (1), (3) 相關的有些結果是我們應該要知道的。
(1)
NOTE. Carathe'odory (metric) outer measure means that for d(A,B)＞0, then
μ^*(A∪B) ＝ μ^*(A) ＋ μ^*(B).
(Theorem) Let μ^* be an outer measure. We have the following equivalent
conditions.
μ^*: metric outer measure
＜＝＞ G in {μ^*-measurable sets} for every open set G.
＜＝＞Ｂ ≦ {μ^*-measurable sets}, where Ｂ is the Borel σ-algebra.
(3)
(Theorem. Hahn Decomposition of X w.r.t. a signed measure)
Let μ be a signed measure, then there exist A ≧ 0, and B ≦ 0 such that
(i) X ＝ A∪B, and (ii) A∩B ＝ ψ.
NOTE.
In general, the decomposition is not unique. For example, say A≧0, and B≦0
and C has measure zero. Then A\C：＝ A' and B∪C：＝ B' is another Hahn de-
composition. However, we can prove the decomposition is unique up to a null
set.
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