Extensions
One may extend the notion of discrete measures to more general measure spaces. Given a measure space and two measures and on it, is said to be discrete in respect to if there exists an at most countable subset of such that
- All singletons with in are measurable (which implies that any subset of is measurable)
Notice that the first two requirements are always satisfied for an at most countable subset of the real line if is the Lebesgue measure, so they were not necessary in the first definition above.
As in the case of measures on the real line, a measure on is discrete in respect to another measure on the same space if and only if has the form
where the singletons are in and their measure is 0.
One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that be zero on all measurable subsets of and be zero on measurable subsets of
Read more about this topic: Discrete Measure
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