exists a measurable function θ with real values such that meaning for any absolutely integrable measurable function f, i.e., f satisfying One can use the Radon–Nikodym theorem to prove that the variation is a measure and the existence of the polar decomposition. The space of complex measures The sum of two complex measures is a complex measure, as is the product of a complex measure by a complex number. That is to say, the set of all complex measures on a measure space (X, Σ) forms a vector space over the complex numbers. Moreover, the total variation defined as is a norm, with respect to which the space of complex measures is a Banach space. See also Riesz representation theorem Signed measure Vector measure External links Complex measure on MathWorld Measures (measure theory)