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I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified.

Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called atom, if there is no proper subset $F\subsetneq E$ such that $F \in \Sigma$.

Alternately:

Version $2$: Let $(X, \Sigma, \mu)$ be a measure space. A set $E ∈ Σ$ is an atom if $\mu(E) > 0$ and whenever $F ∈ Σ, F ⊆ E$ then $\mu(F) = 0$ or $\mu(E \setminus F) = 0$.

The definitions don't seem to be equivalent at least because the first one doesn't even mention measure.

So how should we understand them?

EDIT:

Here is what I've found on MO in the comments to this question:

Pete L. Clark asks:

I know what an atom is in a measure space. What is the definition of an atom in a sigma algebra?

Jonas Meyer responds:

I think it's a nonempty element of the sigma algebra whose only proper subset in the sigma algebra is $∅$.

So the definitions seem to be about different objects. Are they even connected is some way? Does one of them imply another?

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Leo
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  • Related (may be of interest, I don't know exactly if it is to be considered a duplicate): http://math.stackexchange.com/questions/41142/questions-on-atoms-of-a-measure?rq=1 – rubik Apr 21 '14 at 19:45
  • @rubik: I have seen the question yet both definitions there have $\mu$ unlike the first one in my question. – Leo Apr 21 '14 at 19:47

1 Answers1

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In both cases, an atom is a kind of set.

In the first case, an atom is a set in a σ-algebra with a certain property with respect to the σ-algebra. Call this atom_1.

In the second case, an atom is a set in the σ-algebra of a measure with a certain property with respect to the measure. Call this atom_2.

atom_1 is a pointwise concept and doesn't involve a measure. atom_2 is looser because it's still true after modifying by a set of measure zero.

In any case, the two concepts are similar in structure, and can be compared.

Here is a relationship that is easy to see:

An atom_1 in a σ-algebra is necessarily an atom_2 for any measure μ defined on the σ-algebra (provided the atom_1 has positive μ-measure).

But the converse is false because if you add a nonempty set of μ-measure zero to an atom_1 of positive μ-measure it stops being an atom_1 but remains an atom_2.

An atom_1 of zero μ-measure is not an atom_2.

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