In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued linear functional, $\lambda$ defined on the set of continuous functions with compact support in a "nice" topological space (Hausdorff and locally compact) the theorem states there is a corresponding unique positive complete measure $\mu$ defined on a sigma algebra containing all borel sets in X so that the integral of $f$ w.r.t $\mu$, $\int_{X}{f}d\mu$= $\lambda(f)$.
There are a few other parts of the theorem's statement, but I stop here to ask: "What are some examples of corresponding measures to linear functionals that are not immediately obviously translatable to a measure without the theorem?"
An example I thought of first was a linear functional that gives the n'th coefficient of some function $f$ represented by a polynomial, possibly trigonometric. But I also wondered how differentiation at a point would correspond to a measure. Are these questions well-posed? If so could you help me think about them?