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 $ν (E) = ν^{+} (E) - ν^{-} (E) = \lim_{n} (ν^{+} (E_{n}) - ν^{-} (E_{n})) = \lim_{n} ν (E_{n}) .$

1.2L1L3L4

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 $ν (E) = \lim_{n} ν (E_{n})$ .

2.1L1step 1.1

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 $E_{n}$ .

3.1step 1.1step 1.2step 2.1∎

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

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

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)