※ 引述《SMer (想去蒙古騎馬奔馳)》之銘言：
我將原文貼上, 還有不懂的地方以標出, 請教一下清楚的版友 @@
The Hahn Decomposition Theorem
If ν is a siged measure on (X,M), there exist a positive set P and a negative
set N for ν such that P∪N = X and P∩N = ψ. If P', N' is another such pair,
then P△P' (=N△N') is null for ν.
Proof.
W.l.o.g, we assume that ν does not assume the value +oo. (Otherwise, consider
-ν.) We first claim that if ν(A) > -oo, for every ε>0 there exists B < A
such that ν(B) ≧ ν(A) and ν(E) > -ε for all E < B.
If not, there exists E_1 < A with ν(E_1) ≦ -ε.
Since ν(A\E_1) = ν(A) - ν(E_1) ≧ ν(A), there exists E_2 < A\E_1 with
ν(E_2) ≦ -ε. Continuing inductively, we obtain disjoint sets E_j < A
(j in |N) with ν(E_j) ≦ -ε.
oo oo
But then if E = ∪ E_j we have ν(A\E) = ν(A) - Σ ν(E_j) = oo,
j=1 j=1
contrary to assumption. ......
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