Riesz–Markov–Kakutani representation theorem

It relates linear functionals on spaces of continuous functions on a locally compact space to Real or complex measures.

Let $X$ be a locally compact Hausdorff space and $C_c(X)$ be the set of all continuous functions $f:X\to \mathbb{R}$ with compact support.

A positive functional is one that if $f\ge 0$ then $I(f)\ge 0$. An example of positive linear functional is the integral w.r.t. a Radon measure:

$$I(f)={\int }_{X}fd\mu$$

we will show that this is the only possibility.

Note

If $I$ is a positive linear functional on $C_c(X)$, for each compact $K\subset X$ there is a constant $C_K$ such that

$$|I(f)|\le C_K\Vert f{\Vert }_{\infty }\forall f\in C_c(X)\text{ s.t. }\text{supp }f\subseteq K$$

Note X is LCH, it's also locally normal, since we are working in a compact subset K we can use Urysohn’s lemma and build a continuous function \phi \in C_c(X;[0,1]) such that \phi = 1 on K.

Consider real functionals (by the triangle inequality the complex case follows). Since

$$|f|\le \Vert f{\Vert }_{\infty }\varphi$$

or equivalently

$$\Vert f{\Vert }_{\infty }\varphi -f\ge 0,\Vert f{\Vert }_{\infty }\varphi +f\ge 0$$

Since $I$ is a positive linear functional:

$$I(\Vert f{\Vert }_{\infty }\varphi -f)\ge 0,I(\Vert f{\Vert }_{\infty }\varphi +f)\ge 0$$

thus

$$|I(f)|\le I(\varphi )\Vert f{\Vert }_{\infty }\square$$

Note

If $I$ is a positive linear functional on $C_c(X)$, then there is a unique Radon measure $\mu$ on $X$ such that

$$I(f)={\int }_{X}fd\mu ,\forall f\in C_c(X)$$

Moreover,

$$\mu (U)=\sup \{I(f)\mid f\in C_c(X),f\prec U\}$$

for all open sets $U$ and

$$\mu (K)=\inf \{I(f)\mid f\in C_c(X),f\ge 1_K\}$$

for all compact sets $K$.

Note Existence Define for all U open sets:

$$\mu (U):=\sup \{I(f)\mid f\prec U,f\in C_c(X)\}$$

where by $f\prec U$ we mean $0\le f\le 1_U$. We can define an outer measure by:

$${\mu }^{\ast }(E):=\inf \{\mu (U)\mid E\subseteq U,U\text{ open }\}$$

If we restrict it on the Borel set we get a measure. $\dots$