Mapping the delta function and other Radon measures

Consider a continuous function f on the real line with scalar values. It is said to have bounded support if there is a bounded interval [a, b] of real numbers such that f(x) 6= 0 implies a ≤ x ≤ b. An alternative terminology that is often used is that f is said to have compact support. Let Cc denote the vector space of continuous scalar valued functions on the real line, each of which has compact support. This will be called the space of (continuous) test functions. A Radon measure is a positivity preserving linear function μ from Cc to the scalars. The value of the Radon measure μ on the test function f is denoted 〈μ, f〉. The positivity preserving condition says that if for all x we have f(x) ≥ 0, then 〈μ, f〉 ≥ 0. Technical note: In measure theory a Radon measure would generate a measure defined on the Borel subsets of the line that is finite on compact subsets. Example 1. Let h ≥ 0 be a positive locally integrable function. This means that for each point p there is a constant c > 0 such that the integral ∫ p+c