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I don't understand the use of min(t_n/2,1) instead of just t_n, nor the use of max(s_n/2,-1) instead of s_n/2 in the proof of the Hahn decomposition theorem. — Preceding
unsigned comment added by
77.8.166.226 (
talk)
15:41, 7 November 2014 (UTC)reply
Why the sum is ?
The end of the proof says that .
I don't see why it's true.
Each is negative, but they are expected to be increasing because they are infimum of a set which become smaller and smaller since is increasing. What prevents having something like ? In that case whose series converges.
The proof seemed to rely on the assumption that if μ takes on the value -∞, it doesn't take the value +∞. Why is this assumption justifiable?
Kerry (
talk)
16:08, 2 October 2015 (UTC)reply
It is because it would violate the additivity of signed measure. Suppose and are measurable sets with and . Observe that and are mutually disjoint. Consider three cases. (Case: ) We have implying by additivity. Note that the equality can never hold no matter what is. (Case: ) The argument is similar as before. (Case: ) We have implying by additivity. As a result, . Now which is undefined. Since all cases lead to a contradiction, we conclude that a signed measure cannot take both and as values.
Alexvong1995 (
talk)
12:01, 22 December 2018 (UTC)reply