Theorem Riesz Representation [efr-001Z]

Let X be a locally compact Hausdorff topological space, and let C_c(X) denote the vector space of complex-valued, compactly supported functions on X. Recall that a linear functional \psi : C_c \to \mathbb {C} is positive if it carries functions valued in the nonnegative reals to nonnegative real numbers.

For every positive linear functional \psi , there exists a unique Radon measure \mu on X so that \psi (f) = \int _X f(x)d\mu (x) (conversely, it is clear that given a Radon measure, this equation defines a positive linear functional).