TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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Continuity from below, and from above when one set has finite signed measure

Statement

Let $ν$ be a signed measure on $(X, A)$ .

If $E_{0} \subseteq E_{1} \subseteq \dots$ are measurable and $E = ⋃_{n} E_{n}$ , then $ν (E) = \lim_{n \to \infty} ν (E_{n}) .$
If $E_{0} \supseteq E_{1} \supseteq \dots$ are measurable, $E = ⋂_{n} E_{n}$ , and $ν (E_{n_{0}}) \in R$ for some $n_{0}$ , then $ν (E) = \lim_{n \to \infty} ν (E_{n}) .$

Facts & Assumptions

Given: A signed measure $ν$ on $(X, A)$ .

[L1]

Jordan decomposition gives positive measures $ν^{+}, ν^{-}$ with $ν = ν^{+} - ν^{-}$ . (Jordan decomposition of a signed measure into unique mutually singular positive parts)

[L2]

Measures are continuous from below on increasing measurable sequences. (Continuity from below for measures)

[L3]

Measures are continuous from above on decreasing measurable sequences once one term has finite measure. (Continuity from above when one set has finite measure)

[L4]

Every measurable subset of a finite signed-measure set has finite signed measure. (A subset of a set of finite signed measure also has finite signed measure)

Proof

technique · direct

1.1

Let $E_{n} ↑ E$ . By [L1], write $ν = ν^{+} - ν^{-}$ . Then [L2] gives $ν^{+} (E) = \lim_{n} ν^{+} (E_{n}), ν^{-} (E) = \lim_{n} ν^{-} (E_{n}) .$ Subtracting these two equalities yields $ν (E) = ν^{+} (E) - ν^{-} (E) = \lim_{n} (ν^{+} (E_{n}) - ν^{-} (E_{n})) = \lim_{n} ν (E_{n}) .$

1.2

Let $E_{n} ↓ E$ and assume $ν (E_{n_{0}}) \in R$ for some $n_{0}$ . Choose a Hahn decomposition $X = P ⊔ N$ from [L1]. Because $E_{n_{0}} \cap P$ and $E_{n_{0}} \cap N$ are measurable subsets of the finite signed-measure set $E_{n_{0}}$ , [L4] shows that both $ν (E_{n_{0}} \cap P) = ν^{+} (E_{n_{0}})$ and $ν (E_{n_{0}} \cap N) = - ν^{-} (E_{n_{0}})$ are finite. Hence [L3] gives $ν^{+} (E) = \lim_{n} ν^{+} (E_{n}), ν^{-} (E) = \lim_{n} ν^{-} (E_{n}) .$ Subtracting again yields $ν (E) = \lim_{n} ν (E_{n})$ .

L1L3L4

2.1

The displayed subtractions are defined because for a signed measure at most one of $ν^{+} (E)$ and $ν^{-} (E)$ can be infinite, and the same holds for each $E_{n}$ .

L1step 1.1

3.1

Steps 1.1 through 2.1 prove continuity from below and from above under the stated finiteness hypothesis.

step 1.1step 1.2step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Richard F. Bass, Real Analysis for Graduate Students, note after Definition 12.2 (standard reference, not scraped)
John K. Hunter, Measure Theory, §6.6 (standard reference, not scraped)