On the representation of tight functionals as integrals

We consider a vector lattice \(\mathcal L \) of bounded real continuous functions on a topological space \(X\) that separates the points of \(X\) and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on \(\mathcal L \) can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.