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Carathéodory's theorem: measurable sets form a sigma-algebra carrying a complete measure

Statement

For an outer measure $μ^{*}$ on $X$ , the Carathéodory measurable subsets form a sigma-algebra, and the restriction of the outer measure to it is a complete measure.

Facts & Assumptions

Given: An outer measure $μ^{*}$ on $X$ and its family $M_{μ^{*}}$ of Carathéodory measurable sets.

[L1]

The Carathéodory measurable subsets of $X$ form an algebra of subsets. (Carathéodory measurable sets form an algebra)

[L2]

A countable disjoint union of Carathéodory measurable sets is Carathéodory measurable, and $μ^{*} (A \cap ⋃_{k} E_{k}) = \sum_{k} μ^{*} (A \cap E_{k})$ for every $A \subseteq X$ . (Countable disjoint unions of Carathéodory measurable sets are measurable and split every test set)

[L3]

Every set of outer measure zero, and every subset of it, is Carathéodory measurable and has outer measure zero. (Every outer-null set is Carathéodory measurable)

[L4]

Let $A$ be an algebra of subsets of $X$ . If the union of every pairwise disjoint sequence in $A$ belongs to $A$ , then $A$ is a sigma-algebra on $X$ . (An algebra closed under countable disjoint unions is a sigma-algebra)

Proof

technique · direct

1.1

The family $M_{μ^{*}}$ is an algebra by [L1] and is closed under countable disjoint unions by [L2], so [L4] makes it a sigma-algebra.

L1L2L4

1.2

For a pairwise disjoint sequence $(E_{k})$ in $M_{μ^{*}}$ , use $A = ⋃_{k} E_{k}$ in [L2]; then $A \cap E_{k} = E_{k}$ , so $μ^{*} (⋃_{k} E_{k}) = \sum_{k} μ^{*} (E_{k})$ , including infinite values, and the restriction is a measure.

L2algebra

2.1

If $N \in M_{μ^{*}}$ has restricted measure zero and $S \subseteq N$ , then [L3] makes $S$ measurable with outer measure zero; hence the restricted measure is complete.

L3algebra∎

Depends on

Used by

Dependency tree · two levels

15 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)