Signed Measures and Sequences
We have a couple results about signed measures and certain sequences of sets.
If is a signed measure and is a disjoint sequence of measurable sets so that the measure of their disjoint union is finite:
then the series
is absolutely convergent. We already know it converges since the measure of the union is finite, but absolute convergence will give us all sorts of flexibility to reassociate and rearrange our series.
We want to separate out the positive and the negative terms in this series. We write if and otherwise. Similarly, we write if and otherwise. Then we write the two series
The terms of each series have a constant sign — positive for the first and negative for the second — and so if they diverge they can only diverge definitely — to in the first case and to in the second. But at least one must converge or else we’d have obtaining both infinite values. But the sum of all the converges, and so both series must converge — if the series of diverged to and the seris of converged, then there wouldn’t be enough negative terms in the series of for the whole thing to converge. But then since the positive terms and the negative terms both converge, the whole series is absolutely convergent.
Now we turn to some continuity properties. If is a monotone sequence — if it’s decreasing we also ask that at least one — then
The proofs of both of these facts are exactly the same as for measures, except we need the monotonicity result from the end of yesterday’s post to be sure that once we hit one finite , all the later will stay finite.
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