## Singularity

Another relation between signed measures besides absolute continuity — indeed, in a sense the opposite of absolute continuity — is singularity. We say that two signed measures and are “mutually singular” and write if there exists a partition of into two sets so that for every measurable set the intersections and are measurable, and

We sometimes just say that and are singular, or that (despite the symmetry of the definition) “ is singular with respect to “, or vice versa.

In a manner of speaking, if and are mutually singular then all of the sets that give a nonzero value are contained in , while all of the sets that give a nonzero value are contained in , and the two never touch. In contradistinction to absolute continuity, not only does the vanishing of not imply the vanishing of , but if we pare away portions of a set for which gives zero measure then what remains — essentially the only sets for which doesn’t automatically vanish — is necessarily a set for which *does* vanish. Another way to see this is to notice that if and are signed measures with both and , then we must necessarily have ; singularity says that must vanish on any set with , and absolute continuity says must vanish on any set with .

As a quick and easy example, let and be the Jordan decomposition of a signed measure . Then a Hahn decomposition for gives exactly such a partition showing that .

One interesting thing is that singular measures can be added. That is, if and are both singular with respect to , then . Indeed, let and be decompositions showing that and , respectively. That is, for any measurable set we have

Then we can write

It’s easy to check that must vanish on measurable subsets of , and that must vanish on measurable subsets of the remainder of .

[...] As we’ve said before, singularity and absolute continuity are diametrically opposed. And so it’s not entirely surprising that [...]

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