ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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An atomic signed measure on Z has total variation three

Example

On $P (Z)$ , define $ν (A) : = \sum_{k \in A} (- 1)^{k} 2^{- ∣ k ∣} .$ Then $ν$ is a finite signed measure, its positive set is the even integers, its negative set is the odd integers, and $∣ ν ∣ (Z) = \sum_{k \in Z} 2^{- ∣ k ∣} = 3.$

Facts & Assumptions

Given: The set function $ν (A) = \sum_{k \in A} (- 1)^{k} 2^{- ∣ k ∣}$ on $P (Z)$ .

[L1]

Jordan decomposition gives positive and negative parts for a signed measure, and the total variation is their sum. (Jordan decomposition of a signed measure into unique mutually singular positive parts, For a signed measure, total variation is nu-plus plus nu-minus, finite partitions suffice, and nu-plus and nu-minus are extremal)

[A1]

The absolutely summable series $\sum_{k \in Z} 2^{- ∣ k ∣}$ equals $1 + 2 \sum_{n \geq 1} 2^{- n} = 3$ .

Verification

technique · direct

1.1

The defining series is absolutely convergent on every subset of [L1, A1] $Z$ , so $ν$ is countably additive. If $E$ contains only even integers, then every term in the series for $ν (E)$ is nonnegative; if $E$ contains only odd integers, every term is nonpositive. Thus the even integers form a positive set and the odd integers form a negative set.

2.1

The Jordan positive part is therefore the even-atom measure [L1, A1, step 1.1] ∎ $ν^{+} (A) = \sum_{k \in A, k even} 2^{- ∣ k ∣}$ and the negative part is $ν^{-} (A) = \sum_{k \in A, k odd} 2^{- ∣ k ∣}$ . Hence [L1] gives $∣ ν ∣ (Z) = ν^{+} (Z) + ν^{-} (Z) = \sum_{k \in Z} 2^{- ∣ k ∣} = 3$ by [A1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Chapter 12 (standard reference, not scraped)