Riesz Representation Theorem - The Representation Theorem For The Dual of C₀(X)

The Representation Theorem For The Dual of C₀(X)

The following theorem, also referred to as the Riesz-Markov theorem, gives a concrete realisation of the dual space of C₀(X), the set of continuous functions on X which vanish at infinity. The Borel sets in the statement of the theorem also refers to the σ-algebra generated by the open sets.

If μ is a complex-valued countably additive Borel measure, μ is regular iff the non-negative countably additive measure |μ| is regular as defined above.

Theorem. Let X be a locally compact Hausdorff space. For any continuous linear functional ψ on C₀(X), there is a unique regular countably additive complex Borel measure μ on X such that

for all f in C₀(X). The norm of ψ as a linear functional is the total variation of μ, that is

Finally, ψ is positive iff the measure μ is non-negative.

Remark. One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C_c(X) extends in exactly one way to a bounded linear functional on C₀(X), the latter being the closure of C_c(X) in the supremum norm, and that for this reason the first statement implies the second. However the first result is for positive linear functionals, not bounded linear functionals, so the two facts are not equivalent.

In fact, a bounded linear functional on C_c(X) need not remain so if the locally convex topology on C_c(X) is replaced by the supremum norm, the norm of C₀(X). An example is the Lebesgue measure on R, which is bounded C_c(R) but unbounded on C₀(R). This fact can also be seen by observing that the total variation of the Lebesgue measure is infinite.