A real-valued Radon measure is defined to be "any" continuous linear form on formula_2; they are precisely the differences of two Radon measures. This gives an identification of real-valued Radon measures with the dual space of the locally convex space formula_2. These real-valued Radon measures need not be signed measures. For example, sin("x")d"x" is a real-valued Radon measure, but is not even an extended signed measure as it cannot be written as the difference of two measures at least one of which is finite.