A finite additive measure that is continuous from below is a measure
A finite additive measure is a measure iff it is continuous from above. [sorry, I made a mistake in the title, should be "from above"]
I finished the proof from finite additive to continuous from above, but stuck on the other one...
Proof so far.
Suppose that is continuous from above, that is, if with , then I have .
I will show that is a measure.
Let be disjoint sets.
Let , note that and
Now, since is continuous from above, I have
I'm defining the sequence as such base on how I work the other direction, but I can't seem to be able to break the left hand side into a sum of .
Any hints? Thank you