Riesz Representation Theorem - The Representation Theorem For Linear Functionals On C_c(X)

The Representation Theorem For Linear Functionals On C_c(X)

The following theorem represents positive linear functionals on C_c(X), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X. The Borel sets in the following statement refer to the σ-algebra generated by the open sets.

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if

• μ(K) < ∞ for every compact K;
• For every Borel set E,

• The relation

holds whenever E is open or when E is Borel and μ(E) < ∞ .

Theorem. Let X be a locally compact Hausdorff space. For any positive linear functional ψ on C_c(X), there is a unique regular Borel measure μ on X such that

for all f in C_c(X).

One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on C(X). This is the way adopted by Bourbaki; it does of course assume that X starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.

Historical remark: In its original form by F. Riesz (1909) the theorem states that every continuous linear functional A over the space C of continuous functions in the interval can be represented in the form

where α(x) is a function of bounded variation on the interval, and the integral is a Riemann-Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz. (See Gray(1984), for a historical discussion).