194 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Proof: Apply the above lemma to the positive and negative parts of the real and imag-inary parts of f . Let N be the union of the exceptional Borel sets which result. Thus,f XNC is the limit of a sequence hnXNC where hn is continuous and for large enoughn, |hn (x)| ≤ | f (x)| for x /∈ N. Thus hnXNC is Borel and it follows that f XNC is Borelmeasurable. Let g = f XNC . ■

Here is an interesting lemma which is very easy to prove with the above representationtheorem.

Lemma 8.2.12 Suppose µ is a measure defined on the Borel sets of X which is finiteon compact sets. Assume closed balls in X are compact. Then there exists a unique Radonmeasure, µ which equals µ on the Borel sets. In particular µ must be both inner and outerregular on all Borel sets.

Proof: Define a positive linear functional, Λ( f ) =∫

f dµ. Let µ be the Radon measurewhich comes from the Riesz representation theorem for positive linear functionals. Thusfor all f ∈Cc (X) ,

∫f dµ =

∫f dµ. If V is an open set, let { fn} be a sequence of continuous

functions in Cc (X) which is increasing and converges to XV pointwise. Then applying themonotone convergence theorem,∫

XV dµ = µ (V ) =∫

XV dµ = µ (V )

and so the two measures coincide on all open sets. Every compact set is a countable inter-section of open sets and so the two measures coincide on all compact sets. Now let B(a,n)be a ball of radius n and let E be a Borel set contained in this ball. Then by regularity of µ

there exist sets F,G such that G is a countable intersection of open sets and F is a countableunion of compact sets such that F ⊆ E ⊆ G and µ (G\F) = 0. Now µ (G) = µ (G) andµ (F) = µ (F) . Thus

µ (G\F)+µ (F) = µ (G) = µ (G) = µ (G\F)+µ (F)

and so µ (G\F) = µ (G\F) = 0. It follows µ (E) = µ (F) = µ (F) = µ (G) = µ (E) . IfE is an arbitrary Borel set, then µ (E ∩B(a,n)) = µ (E ∩B(a,n)) and letting n→ ∞, thisyields µ (E) = µ (E) . ■

8.3 Approximation with Gδ and Fσ

The inner and outer regularity results imply an important Proposition which is partly al-luded to in the above.

Definition 8.3.1 A countable union of closed sets is called an Fσ set and a count-able intersection of open sets is called a Gδ set. Obviously these sets are Borel sets.

Proposition 8.3.2 Let µ be the Radon measure from Theorem 8.2.1 coming from apositive linear functional on Cc (X) for X a metric space in which closed balls are compact,and let F be the σ algebra obtained there. Then if E ∈F , there exists F an Fσ set andG a Gδ set such that F ⊆ E ⊆ G and µ (F) = µ (E) = µ (G). It can also be assumedthat µ (G\F) = 0. If f ∈ L1 (X ,F ,µ) , then there exists g ∈ L1 (X ,B (X) ,µ) such that|g(x)| ≤ | f (x)| and g(x) = f (x) off a Borel set of measure zero.