The set of all $\mu^*$-measurable sets is a ring

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Statement

$\mathcal{S}$, the set of all $\mu^*$ measurable sets, is a ring of sets.

Recall that given an outer-measure, $\mu^*:H\rightarrow\bar{\mathbb{R} }_{\ge 0}$, where $H$ is a hereditary $\sigma$-ring that we call a set, $A\in H$ $\mu^*$-measurable if:

• $\forall B\in H[\mu^*(B)=\mu^*(B\cap A)+\mu^*(B-A)]$.
• See the page $\mu^*$-measurable set for more information.

Proof

[Expand]Recall what we must show in order for $\mathcal{S}$ to be a ring of sets:

It is sufficient to show only:

• $\forall A,B\in \mathcal{S}[A\cup B\in \mathcal{S}]$
• $\forall A,B\in \mathcal{S}[A-B\in \mathcal{S}]$

(Unknown grade)

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Extremely well documented in Halmos

See also

• The set of all $\mu^*$-measurable sets is a $\sigma$-ring
• The restriction of an outer-measure to the set of all $\mu^*$-measurable sets is a measure