real analysis – Risk representation for ring homomorphism

Leave $ X $ Be a compact metric space. Prove that any homomorphism ring $ C (X) a mathbb {R} $ it's an evaluation $ f to f (x_0) $ for some point $ x_0 in X $.

$ textbf {Question} $:

Have the test sketch as described below. but I'm not sure of step 3 and I also have problems with the final step.

$ textbf {Test sketch} $:

1) Show that such homomorphism is a positive linear functional.

2) Apply the Riesz representation theorem to prove that it is
given by integration with respect to some measure.

3) Use the homomorphism property to argue that if U, V are disjoint open sets, at most, one can have a positive measure.

4) Deduce that the measure in question is a delta measure of Dirac at some point.

$ textbf {My attempt} $:

leave $ Lambda: C (X) to mathbb {R} $, then assuming it is a ring homomorphism and has the following properties for $ f, g in C (X) $; we have that $ (i) $ $ Lambda (f + g) = Lambda (f) + Lambda (g) $ ,$ (ii) $ $ Lambda (f.g) = Lambda (f). Lambda (g) $ Y $ (iii) $ $ Lambda (1_ {X}) = 1 $ or $ Lambda (0_X) = 0 $

$ textit {Step 1}: $ I need to prove that yes $ f geq0 $ so $ Lambda (f) geq0 $.

leave $ f geq0 $ be any continuous function, we can approximate that by simple functions $ f = lim_ {n to infty} S_n $ . Applying the operator $ Lambda (f) = lim_ {n to infty} Lambda (S_n) $. Then it is enough to show that

$ A) $ $ Lambda (S) geq0 $ for any simple function $ S = sum_ {i = 1} ^ infty a_i mathcal {X_ {A_ {i}}} $ , $ a_i geq 0 $

$ B) $ $ Lambda ( mathcal {X_ {A}}) geq0 $

i will show $ B $ so $ A $.

leave $ {f_n } _ {n = 1} ^ infty $ be a sequence of continuous functions that we approach the characteristic function (since it is not continuous) by them. Then I start with a characteristic function $ lim_ {n to infty} f_n ^ A = mathcal {X_A} $.
The clearly characteristic function satisfies the above $ (i) – (iii) $ and clearly since it takes zero or one, it is positive.

To show A we can write that by $ (i) $ , $ Lambda (S) = sum_ {i = 1} ^ infty a_i Lambda ( mathcal {X_ {A_ {i}})} geq 0 $ how $ a_i geq0 $

$ textit {Step 2}: $ Then from $ Lambda $ is a positive linear operator by risk representation we have to $ exists $ measure $ nu $ that we write the functional as $ Lambda (f) = int_X fd nu $

$ textit {Step 3}: $ Leave $ U, V subset X $ be disjoint Then we can write for any of the two characteristic functions (which we would assume are the limit of continuous functions)

$ Lambda ( mathcal {X} _U). Lambda ( mathcal {X} _V) = $ 0 Then, at most, one of them can be positive. We also have that

$ Lambda ( mathcal {X} _U) = int_X mathcal {X} _U d nu = int_U d nu = nu (U) $
so $ nu (U). nu (V) = 0 $ which implies that at most one of them can be positive.

$ textit {Step 4}: $