156 1. Measure theory (iii) Show that the Hahn-Kolmogorov extension μ: B[R] → [0, +∞] of μ0 assigns an infinite measure to any non-empty Borel set. (iv) Show that counting measure # (or more generally, c# for any c ∈ (0, +∞]) is another extension of μ0 on B[R]. Exercise 1.7.9. Let μ0 : B0 → [0, +∞] be a pre-measure which is σ-finite (thus X is the countable union of sets in B0 of finite μ0-measure), and let μ: B → [0, +∞] be the Hahn-Kolmogorov extension of μ0. (i) Show that if E ∈ B, then there exists F ∈ B0 containing E such that μ(F \E) = 0 (thus F consists of the union of E and a null set). Furthermore, show that F can be chosen to be a countable intersection F = ∞ n=1 Fn of sets Fn, each of which is a countable union Fn = ∞ m=1 Fn,m of sets Fn,m in B0. (ii) If E ∈ B has finite measure (i.e. μ(E) ∞), and ε 0, show that there exists F ∈ B0 such that μ(EΔF ) ≤ ε. (iii) Conversely, if E is a set such that for every ε 0 there exists F ∈ B0 such that μ∗(EΔF ) ≤ ε, show that E ∈ B. 1.7.3. Lebesgue-Stieltjes measure. Now we use the Hahn-Kolmogorov extension theorem to construct a variety of measures. We begin with Lebesgue-Stieltjes measure. Theorem 1.7.9 (Existence of Lebesgue-Stieltjes measure). Let F : R → R be a monotone non-decreasing function, and define the left and right limits F−(x) := sup yx F (y), F+(x) := inf yx F (y) thus one has F−(x) ≤ F (x) ≤ F+(x) for all x. Let B[R] be the Borel σ- algebra on R. Then there exists a unique Borel measure μF : B[R] → [0, +∞] such that μF ([a, b]) = F+(b) − F−(a),μF ([a, b)) = F−(b) − F−(a), (1.33) μF ((a, b]) = F+(b) − F+(a),μF ((a, b)) = F−(b) − F+(a), for all −∞ b a ∞, and (1.34) μF ({a}) = F+(a) − F−(a) for all a ∈ R. Proof. (Sketch) For this proof, we will deviate from our previous nota- tional conventions, and allow intervals to be unbounded, thus, in particular, including the half-infinite intervals [a, +∞), (a, +∞), (−∞,a], (−∞,a) and the doubly infinite interval (−∞, +∞) as intervals. Define the F -volume |I|F ∈ [0, +∞] of any interval I, adopting the obvi- ous conventions that F−(+∞) = supy∈R F (y) and F+(−∞) = infy∈R F (y),

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