Where to Find Proofs of the Riesz–Markov–Kakutani Theorem

The Riesz–Markov–Kakutani theorem connects positive linear functionals on spaces of continuous functions with measures. There are several closely related versions, so a proof that is right for one setting may not apply unchanged to another. This guide states the common versions, sketches the standard construction, and points to reliable textbook proofs.

The usual versions

Let $K$ be a compact Hausdorff space. For the real version, let $C(K)$ denote the continuous real-valued functions on $K$, and let $L:C(K)\to\mathbb{R}$ be a positive real-linear functional. For the complex version, let $C(K)$ denote the continuous complex-valued functions on $K$, and let $L:C(K)\to\mathbb{C}$ be a positive complex-linear functional, meaning that $L(f)\geq 0$ whenever $f$ is real-valued and $f\geq 0$. In either case, there is a unique finite regular Borel measure $\mu$ such that

$$L(f)=\int_K f\,d\mu\qquad(f\in C(K)).$$

For this positive version, $\mu(K)=L(1)$ and the operator norm satisfies $\|L\|=L(1)$. The theorem also has a locally compact Hausdorff version: bounded positive linear functionals on $C_0(X)$ correspond to finite regular Borel measures on $X$. One can formulate a related result for $C_c(X)$, the compactly supported continuous functions, with the appropriate local boundedness condition; the representing measure is then generally only locally finite.

If the functional is not positive, versions of the theorem represent bounded linear functionals by signed or complex measures. The positive case is the core result; the general case follows by decomposing a real functional into positive parts and treating real and imaginary parts separately.

The standard construction in outline

For the compact-space version, start with a positive $L$ on $C(K)$. For each open set $U\subseteq K$, define

$$\mu(U)=\sup\{L(f): f\in C(K),\ 0\leq f\leq 1,\ \operatorname{supp}(f)\subseteq U\}.$$

Here $\operatorname{supp}(f)$ is the closure of the set where $f$ is nonzero. This definition asks how much of the functional can be detected by continuous functions that live inside $U$. Positivity gives monotonicity: if $U\subseteq V$, then $\mu(U)\leq\mu(V)$.

The next steps turn these open-set values into a measure. Define an outer measure by covering an arbitrary set with open sets and taking the infimum of the sums of their $\mu$-values. The key technical work is proving the necessary subadditivity and showing that Borel sets are measurable for this outer measure. Compactness and the existence of continuous functions separating compact sets from closed sets supply the approximation and covering arguments.

The resulting Borel measure is regular. One then proves that its integral agrees with $L$ first for suitable nonnegative continuous functions and, by approximation and linearity, for every $f\in C(K)$. The open-set formula also yields uniqueness: any representing regular measure must have the same values on open sets, and regularity then determines its values on Borel sets.

This is a proof roadmap rather than a substitute for the technical details. In a full proof, the construction of the outer measure and the integral approximation are the main places to check carefully.

A concrete example

On $K=[0,1]$, define

$$L(f)=2f(0)+\int_0^1 f(x)\,dx.$$

This is linear and positive: if $f(x)\geq 0$ everywhere, both terms are nonnegative. Its representing measure is

$$\mu=2\delta_0+\lambda,$$

where $\delta_0$ is unit mass at $0$ and $\lambda$ is Lebesgue measure restricted to $[0,1]$. Indeed,

$$\int_{[0,1]} f\,d\mu=2f(0)+\int_0^1 f(x)\,dx=L(f).$$

The total mass is $\mu([0,1])=2+1=3$, which agrees with $L(1)=3$ and, by positivity, with $\|L\|$.

Where to read full proofs

Two standard references are:

• Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition. See the treatment of the Riesz representation theorem and regular Borel measures.
• Walter Rudin, Real and Complex Analysis, 3rd edition. Its discussion of positive linear functionals and measure representation gives a classical route through the theorem.

When comparing proofs, first check which function space and hypotheses the author uses. A proof for compact metric spaces may use sequences or metric regularity arguments that do not directly handle arbitrary compact Hausdorff spaces. A proof for $C_0(X)$ instead needs the locally compact setting and an account of how functions vanishing at infinity are handled.

Common mistakes

• Dropping positivity: Positivity is a hypothesis for the direct positive-measure statement. A general bounded functional is represented by a signed or complex measure, not necessarily a positive one.
• Confusing $C(K)$ with $C_0(X)$: On a noncompact space, the constant function $1$ may not belong to $C_0(X)$. Consequently, the compact-space identity $\mu(K)=L(1)$ does not transfer as written.
• Ignoring regularity: The theorem specifies a regular Borel measure. Merely producing a measure with the right integrals does not complete the usual uniqueness argument unless the relevant regularity conditions are addressed.
• Assuming the space is metric: Compact Hausdorff spaces need not be metrizable. Metric-only steps in a proof require additional hypotheses or a replacement argument.

For the compact-metric special case, proofs can often be made more concrete using approximation on compact sets. For the general compact Hausdorff theorem, look for a proof that explicitly uses topological separation and regularity rather than sequences alone.