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Context

The notion of atoms and point masses agree to certain extent.
(See Summary on Atoms.)

Measures decompose w.r.t. atoms.
(See Paper on Atoms.)

Here, the goal is a direct approach to decompose w.r.t. point masses!

Problem

Consider a sigma-finite measure $\mu:\Sigma\to\mathbb{R}_+$.

Does it decompose into a discrete and a continuous part: $$\mu=\mu_0+\mu_\infty$$ (For a precise definition see corresponding paragraph of Summary on Atoms.)

Attempt

My sort of dumb idea is to simply subtract all discrete masses: $$\mu_\infty:=\mu-\sum_{\#D\leq\aleph_0}\mu(D)$$ (However, there are a loooot of double countings...)

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freishahiri
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1 Answers1

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Decomposition

Denote the collection of atoms by: $$\mathcal{A}:=\{A\in\Sigma:A\text{ atom}\}$$

Identify common atoms: $$A\sim A':\iff A\cap A'\in\mathcal{A}$$

Collect representatives: $$A:=\bigcup_{[A]\in\mathcal{A}/\sim}A$$ (By sigma-finiteness there can be at most countably many!)

Construct atomic and atomless part: $$\mu_a(E):=\mu(E\cap A)$$ $$\mu_c(E):=\mu(E\cap A)$$

Remark

For set atoms and point masses the proof follows the same lines!

freishahiri
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