Complex measure

Summary

A complex measure is a countably additive function from a σ-algebra to the complex numbers. Its real and imaginary parts are finite signed measures. Complex measures form a Banach space under the total variation norm. On a locally compact Hausdorff space, the Riesz–Markov–Kakutani theorem identifies the complex Radon measures with the dual of the continuous functions vanishing at infinity. They are the natural measures for Fourier analysis, where one integrates complex exponentials.