A polar representation of a complex measure μ is analogous to the polar representation of a complex number as z = r e^(i θ), where r = left bracketing bar z right bracketing bar , d μ = e^(i θ) d left bracketing bar μ right bracketing bar . The analog of absolute value is the total variation left bracketing bar μ right bracketing bar , and θ is replaced by a measurable real-valued function θ. Or sometimes one writes h with left bracketing bar h right bracketing bar = 1 instead of e^(i θ). More precisely, for any measurable set E, μ(E) = integral_E e^(i θ) d left bracketing bar μ right bracketing bar , where the integral is the Lebesgue integral.

absolutely continuous | complex measure | fundamental theorems of calculus | Lebesgue measure | phasor | Radon-Nikodym theorem