Definition 10.3.1 Let (Ω,τ) be a topological space. L : C_{c}(Ω) → ℂ is called a positive linear functional if L is linear,

and if Lf ≥ 0 whenever f ≥ 0.
Theorem 10.3.2 (Riesz representation theorem) Let (Ω,τ) be a locally compact Hausdorff space and let L be a positive linear functional on C_{c}(Ω). Then there exists a σ algebra S containing the Borel sets and a unique measure μ, defined on S, such that

for all F open and for all F ∈S with μ(F) < ∞,

for all F ∈S, and
 (10.3.11) 
The plan is to define an outer measure and then to show that it, together with the σ algebra of sets measurable in the sense of Caratheodory, satisfies the conclusions of the theorem. Always, K will be a compact set and V will be an open set.
Definition 10.3.3 μ(V ) ≡ sup{Lf : f ≺ V } for V open, μ(∅) = 0.μ(E) ≡ inf{μ(V ) : V ⊇ E} for arbitrary sets E.
Proof: First it is necessary to verify μ is well defined because there are two descriptions of it on open sets. Suppose then that μ_{1}
It remains to show that μ is an outer measure. Let V = ∪_{i=1}^{∞}V _{i} and let f ≺ V . Then spt(f) ⊆∪_{i=1}^{n}V _{i} for some n. Let ψ_{i} ≺ V _{i},∑ _{i=1}^{n}ψ_{i} = 1 on spt(f).

Hence

since f ≺ V is arbitrary. Now let E = ∪_{i=1}^{∞}E_{i}. Is μ(E) ≤∑ _{i=1}^{∞}μ(E_{i})? Without loss of generality, it can be assumed μ(E_{i}) < ∞ for each i since if not so, there is nothing to prove. Let V _{i} ⊇ E_{i} with μ(E_{i}) + ε2^{−i} > μ(V _{i}).

Since ε was arbitrary, μ(E) ≤∑ _{i=1}^{∞}μ(E_{i}) which proves the lemma.
Lemma 10.3.5 Let K be compact, g ≥ 0,g ∈ C_{c}(Ω), and g = 1 on K. Then μ(K) ≤ Lg. Also μ(K) < ∞ whenever K is compact.
Proof: Let α ∈ (0,1) and V _{α} = {x : g(x) > α} so V _{α} ⊇ K and let h ≺ V _{α}.
Then h ≤ 1 onV _{α} while gα^{−1} ≥ 1 on V _{α}and so gα^{−1} ≥ h which implies L(gα^{−1}) ≥ Lh and that therefore, since L is linear,

Since h ≺ V _{α} is arbitrary, and K ⊆ V _{α},

Letting α ↑ 1 yields Lg ≥ μ(K). This proves the first part of the lemma. The second assertion follows from this and Theorem 10.2.7. If K is given, let

and so from what was just shown, μ
Proof: By Theorem 10.2.7 or 10.2.8, there exists h ∈ C_{c}
From Lemma 10.3.5 μ(A ∪ B) < ∞ and so there exists an open set, W such that

Now let U = U_{1} ∩ W and V = V _{1} ∩ W. Then

Let A ≺ f ≺ U,B ≺ g ≺ V . Then by Lemma 10.3.5,

Since ε > 0 is arbitrary, this proves the lemma.
From Lemma 10.3.5 the following lemma is obtained.
Proof: Let V ⊇ spt(f) and let spt(f) ≺ g ≺ V . Then Lf ≤ Lg ≤ μ(V ) because f ≤ g. Since this holds for all V ⊇ spt(f),Lf ≤ μ(spt(f)) by definition of μ.
Finally, let V be open and let l < μ
At this point, the conditions of Lemma 10.1.6 have been verified. Thus S contains the Borel sets and μ is inner regular on sets of S having finite measure.
It remains to show μ satisfies 10.3.11.
Proof: Let f ∈ C_{c}(Ω),f realvalued, and suppose f(Ω) ⊆ [a,b]. Choose t_{0} < a and let t_{0} < t_{1} <
 (10.3.12) 
Note that ∪_{i=1}^{n}E_{i} is a closed set, and in fact
 (10.3.13) 
since Ω = ∪_{i=1}^{n}f^{−1}((t_{i−1},t_{i}]). Let V _{i} ⊇ E_{i},V _{i} is open and let V _{i} satisfy
 (10.3.14) 

By Theorem 10.2.11 there exists h_{i} ∈ C_{c}(Ω) such that

Now note that for each i,

(If x ∈ V _{i}, this follows from 10.3.14. If x




From 10.3.13 and 10.3.12, the first and last terms cancel. Therefore this is no larger than

Since ε > 0 is arbitrary,
 (10.3.15) 
for all f ∈ C_{c}(Ω),f real. Hence equality holds in 10.3.15 because L(−f) ≤−∫ fdμ so L(f) ≥∫ fdμ. Thus Lf = ∫ fdμ for all f ∈ C_{c}(Ω). Just apply the result for real functions to the real and imaginary parts of f. This proves the Lemma.
This gives the existence part of the Riesz representation theorem.
It only remains to prove uniqueness. Suppose both μ_{1} and μ_{2} are measures on S satisfying the conclusions of the theorem. Then if K is compact and V ⊇ K, let K ≺ f ≺ V . Then

Thus μ_{1}(K) ≤ μ_{2}(K) for all K. Similarly, the inequality can be reversed and so it follows the two measures are equal on compact sets. By the assumption of inner regularity on open sets, the two measures are also equal on all open sets. By outer regularity, they are equal on all sets of S. This proves the theorem.
An important example of a locally compact Hausdorff space is any metric space in which the closures of balls are compact. For example, ℝ^{n} with the usual metric is an example of this. Not surprisingly, more can be said in this important special case.
Theorem 10.3.10 Let
Proof: Let μ and S be as described in Theorem 10.3.2. The outer regularity comes automatically as a conclusion of Theorem 10.3.2. It remains to verify inner regularity. Let F ∈S and let l < k < μ

Since μ

This proves inner regularity. In case
The proof of the above yields the following corollary.
The following is on the uniqueness of the σ algebra in some cases.
Definition 10.3.12 Let
Corollary 10.3.13 Let

and let L be a positive linear functional defined on C_{c}
Proof: Suppose
