A complex measure is a finite-valued countably additive set function

Depends on

• Sigma-algebras
• $\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$

Used by

• A positive, signed, or complex measure concentrated on a measurable set Definition
• Absolute continuity of a signed or complex measure with respect to a positive measure Definition
• Fourier restriction and adjoint extension operators Definition
• Mutual singularity for signed or complex measures Definition
• Regular complex Borel measures Definition
• The simple integral against a signed or complex measure Definition
• The total variation |nu|(E) from countable measurable partitions Definition
• A boundary atom gives an h1 function without an L1 density Example
• Complex circle measures have finite regular total variation under countable choice Lemma
• Continuous functional calculus produces a regular PVM Lemma
• Fourier pairing for a finite measure and Schwartz data Lemma
• Scalar and complex measures from a pvm Lemma
• TT-star reduces extension to convolution with the surface-measure transform Lemma
• The real and imaginary parts of a complex measure are finite signed measures, and nu = Re nu + i Im nu Proposition
• A complex L¹ density defines a complex measure whose total variation is |h| dmu Theorem
• Every complex measure has finite total variation Theorem
• Fourier transform of a finite complex Borel measure Theorem
• Positive harmonic boundary measures and compact normalized families Theorem

Dependency tree · two levels

8 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

• John K. Hunter, Measure Theory, Definition 6.29 (standard reference, not scraped)
• Richard F. Bass, Real Analysis for Graduate Students, Exercise 13.3 (standard reference, not scraped)