Riesz-Markov-Kakutani representation theorem

Let $X$ be a locally compact Hausdorff space and let $C_c(X)$ be the space of continuous real functions of compact support. For every positive linear functional $I$ on $C_c(X)$, that is, every linear $I$ with $I(f)\ge 0$ whenever $f\ge 0$, there is a unique Radon measure $\mu$ on the Borel sets of $X$ with

$I(f)={\int }_{X}fd\mu \text{for all }f\in C_c(X)\mathrm{.}$

Here Radon means: $\mu (K)<\mathrm{\infty }$ for every compact $K$, $\mu$ is outer regular on all Borel sets, and $\mu$ is inner regular on open sets and on Borel sets of finite measure.

Dual form. For such an $X$, the dual of $C_0(X)$ with the supremum norm is isometrically isomorphic to the space of regular complex Borel measures on $X$ with the total variation norm; the pairing is integration, and $\mathrm{\parallel }\psi \mathrm{\parallel }=\mathrm{\mid }\mu \mathrm{\mid }(X)$. For compact Hausdorff $X$ this identifies $C(X{)}^{\ast }$.

The two forms are not the same statement and are constantly conflated. The first is about positive functionals on compactly supported functions and asserts a positive measure; the second is about all bounded functionals on $C_0(X)$ and asserts a complex measure of finite total variation. The first needs no boundedness hypothesis, since positivity already forces local boundedness; the second needs no positivity.

Not proved in this library, and doubly deferred. It needs the functional analysis track for the duality statement and the measure and integration track (Lebesgue measure and the Lebesgue integral ‡) for the measures themselves, and for the integral in which the conclusion is written. Both tracks are recorded as missing.

What would prove it. A Caratheodory style construction: define an outer measure from the functional by taking infima of $I(f)$ over functions dominating the indicator of an open set, verify regularity, and check that integration against the resulting measure reproduces the functional on $C_c(X)$. The regularity hypothesis is not decoration: without it uniqueness fails, since on a badly behaved $X$ distinct Borel measures can integrate every $f\in C_c(X)$ to the same value.

Why it matters here. It is the theorem that makes measure theory and functional analysis two descriptions of one subject, by identifying a purely analytic object, a positive functional on continuous functions, with a purely measure-theoretic one. It belongs beside Lusin's theorem (Lusin's theorem ‡), which is the same fact read in the other direction: measurable behaviour is continuous behaviour off a small set, and continuous functions are therefore enough to see the measure. It is also the concrete half of the algebra and topology dictionary: together with the Banach-Stone theorem and the commutative Gelfand-Naimark theorem it says that the compact Hausdorff space $X$, the Banach space $C(X)$, the algebra $C(X)$ and the measures on $X$ are four views of the same data.