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A sigma-finite premeasure has at most one extension to its generated sigma-algebra

Statement

A sigma-finite premeasure has at most one measure extension to the sigma-algebra generated by its source algebra.

Facts & Assumptions

Given: A sigma-finite premeasure $μ_{0}$ on $A_{0}$ , a sequence $(A_{n})$ in $A_{0}$ covering $X$ with finite premeasure, and extensions $μ, ν$ on $σ (A_{0})$ .

[F1]

A premeasure on an algebra $A_{0}$ vanishes at the empty set and is countably additive whenever a disjoint sequence in $A_{0}$ has its union in $A_{0}$ . (Premeasures on algebras of sets)

[F2]

An algebra of subsets is nonempty and is closed under finite unions and intersections. (Algebras of subsets)

[F3]

A pi-system on $X$ is a nonempty family of subsets closed under binary intersections. (Pi-systems)

[L1]

If two measures agree on a generating pi-system and on an increasing exhaustion $(P_{n})$ from that pi-system with $X = ⋃_{n} P_{n}$ and equal finite values on every $P_{n}$ , then the measures are equal on the generated sigma-algebra. (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system)

Proof

technique · direct

1.1

Put $P_{n} = ⋃_{k \leq n} A_{k}$ . By [F2], $P_{n} \in A_{0}$ , the sequence is increasing with union $X$ , and finite additivity from [F1] gives $μ_{0} (P_{n}) \leq \sum_{k \leq n} μ_{0} (A_{k}) < + \infty$ .

F1F2algebra

2.1

The algebra $A_{0}$ is nonempty and closed under intersections by [F2], hence is a pi-system by [F3]; it generates $σ (A_{0})$ , and both extensions agree with $μ_{0}$ on it and on the exhaustion $(P_{n})$ from step 1.1.

step 1.1F3given

3.1

Applying [L1] to the generating pi-system and the increasing finite-measure exhaustion of step 2.1 gives $μ = ν$ on $σ (A_{0})$ .

step 2.1L1∎

Depends on

Used by

A finite premeasure has at most one extension to its generated sigma-algebra Corollary

Dependency tree · two levels

12 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 1.14 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory, Exercise 1.7.7 (standard reference, not scraped)