Definition: Let ( X , <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="script">A

Thomas Hubbard

Thomas Hubbard

Answered question

2022-05-21

Definition:
Let ( X , A , μ ) be a measurable space, an atom of the measure μ is a set A A with the property that μ ( A ) > 0 and for any B σ ( A ) either μ ( B ) = 0, or μ ( A B ) = 0. If a measure has atoms it is called atomic; in the opposite case, the measure is called non-atomic (or atomeless). A measure is called purely atomic if X can be written as the union of a finite or countable number of atoms.
From the definition of atoms, we get the following corollary:
Corollary:
Every purely atomic measure is an atomic measure.
I am trying to find an example of an atomic measure that is not purely atomic, can anyone help me?

Answer & Explanation

Pedro Hurley

Pedro Hurley

Beginner2022-05-22Added 9 answers

The sum of the Dirac measure and the Lebesgue measure in R is atomic but not purely atomic, because { 0 } is its only atom.

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