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Carathéodory measurable sets form an algebra

Statement

The Carathéodory measurable subsets of $X$ form an algebra of subsets. In particular, $\emptyset$ is measurable, complements of measurable sets are measurable, and finite unions of measurable sets are measurable (Algebras of subsets).

Facts & Assumptions

Given: An outer measure $μ^{*}$ on $X$ and Carathéodory measurable sets $E, F \subseteq X$ .

[F1]

A set $E \subseteq X$ is Carathéodory measurable for $μ^{*}$ when $μ^{*} (A) = μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ for every $A \subseteq X$ . (Carathéodory measurable sets)

[L1]

For every outer measure $μ^{*}$ and all $A, E \subseteq X$ , $μ^{*} (A) \leq μ^{*} (A \cap E) + μ^{*} (A ∖ E)$ . (In the Carathéodory identity, the subadditive inequality is automatic)

Proof

technique · direct

1.1

For every test set $A$ , the split by $\emptyset$ reads $μ^{*} (A) = 0 + μ^{*} (A)$ , so $\emptyset$ is measurable; the identity for $X ∖ E$ is the identity for $E$ with its two summands exchanged. Applying [F1] first to $E$ and then, inside each resulting piece, to $F$ splits $A$ into the four cells $A \cap E \cap F$ , $A \cap E ∖ F$ , $A \cap F ∖ E$ , and $A ∖ (E \cup F)$ , with their outer measures summing to $μ^{*} (A)$ .

F1algebra

2.1

By subadditivity, $μ^{*} (A \cap (E \cup F))$ is at most the sum of the first three cell values from step 1.1, while $A ∖ (E \cup F)$ is the fourth cell; hence $μ^{*} (A) \geq μ^{*} (A \cap (E \cup F)) + μ^{*} (A ∖ (E \cup F))$ . The reverse inequality is [L1], so $E \cup F$ is measurable and the measurable family is an algebra.

step 1.1F1L1algebra∎

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Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. Folland, Real Analysis, 2nd ed., Theorem 1.11 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory, Theorem 1.7.3 (standard reference, not scraped)