In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory.
The theorem is named for Frigyes Riesz (1909) who introduced it for continuous functions on the unit interval, Andrey Markov (1938) who extended the result to some non-compact spaces, and Shizuo Kakutani (1941) who extended the result to compact Hausdorff spaces.
The statement of the theorem for positive linear functionals on Cc(X), the space of compactly supported complex-valued continuous functions, is as follows: Theorem Let X be a locally compact Hausdorff space and
Then there exists a unique positive Borel measure
containing the Borel σ-algebra on X: As such, if all open sets in X are σ-compact then
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.
Without the condition of regularity the Borel measure need not be unique.
The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω.
The following representation, also referred to as the Riesz–Markov theorem, gives a concrete realisation of the topological dual space of C0(X), the set of continuous functions on X which vanish at infinity.
Theorem Let X be a locally compact Hausdorff space.
on C0(X), there is a unique complex-valued regular Borel measure
In its original form by Frigyes Riesz (1909) the theorem states that every continuous linear functional A over the space C([0, 1]) of continuous functions f in the interval [0, 1] can be represented as where α(x) is a function of bounded variation on the interval [0, 1], 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.