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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The complex density e^{ix} dlambda has total variation 2pi

Example

On $B ([0, 2 π])$ , define $ν (E) : = \int_{E} e^{i x} d λ .$ Then $ν$ is a complex measure and $∣ ν ∣ ([0, 2 π]) = \int_{0}^{2 π} ∣ e^{i x} ∣ d x = 2 π .$

Facts & Assumptions

Given: The density $h (x) = e^{i x}$ on $[0, 2 π]$ .

[L1]

A complex $L^{1}$ density $h$ defines a complex measure with $∣ ν ∣ (E) = \int_{E} ∣ h ∣ d μ$ . (A complex L^1 density defines a complex measure whose total variation is |h| dmu)

[A1]

One has $∣ e^{i x} ∣ = 1$ for every real $x$ .

Verification

technique · direct

1.1

The function $h (x) = e^{i x}$ is bounded on the finite interval $[0, 2 π]$ , [L1, A1] so it lies in $L^{1} (λ)$ . Therefore [L1] makes $ν (E) = \int_{E} e^{i x} d λ$ a complex measure.

2.1

Applying [L1] and then [A1] on the whole interval gives [L1, A1, step 1.1] ∎ $∣ ν ∣ ([0, 2 π]) = \int_{0}^{2 π} ∣ e^{i x} ∣ d x = \int_{0}^{2 π} 1 d x = 2 π .$

Depends on

A complex L^1 density defines a complex measure whose total variation is |h| dmu

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, Exercise 13.4 (standard reference, not scraped)