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 Incoming links: Maximal additivity algebra of a regular outer measure

E \subseteq X

, a measurable hull

H \supseteq E

with

μ (H) = μ (E)

Suppose

C

is an algebra containing

M_{μ}^{*}

and

μ

is finitely additive on

C

. Fix

A \in C

and an arbitrary

E

of finite outer measure. Take a measurable hull

H

E

. Both

H \cap A

and

H ∖ A

belong to

C

, so

μ (E) = μ (H) = μ (H \cap A) + μ (H ∖ A) \geq μ (E \cap A) + μ (E ∖ A) .

Outer subadditivity gives equality. If

μ (E) = \infty

, outer subadditivity already forces the sum on the right to be infinite, again giving equality. Therefore

A

satisfies the Caratheodory criterion. We have proved

C \subseteq M_{μ}^{*}

, so

C = M_{μ}^{*} .