Multivariable Advanced Calculus

164 MEASURES AND MEASURABLE FUNCTIONSThis establishes µ is regular if µ is finite.Now suppose it is only known that µ is finite on compact sets. Consider outerregularity. There are at most finitely many r ∈ [0, R] such that µ (S (0,r)) > δ >0. If this were not so, then µ (D (0,R)) = ∞ contrary to the assumption that µ isfinite on compact sets. Therefore, there are at most countably many r ∈ [0, R] suchthat µ (S (0,r)) > 0. Here is why. Let S k denote those values of r ∈ [0, R] such thatµ (S (0,r)) > 1/k. Then the values of r such that µ (S (0,r)) > 0 equals ∪ ∞ m=1S m , acountable union of finite sets which is at most countable.It follows there are at most countably many r ∈ (0, ∞) such that µ (S (0,r)) > 0.Therefore, there exists an increasing sequence {r k } such that lim k→∞ r k = ∞ andµ (S (0,r k )) = 0. This is easy to see by noting that (n, n+1] contains uncountably manypoints and so it contains at least one r such that µ (S (0,r)) = 0.S (0,r) = ∩ ∞ k=1 (B (0, r + 1/k) − D (0,r − 1/k))a countable intersection of open sets which are decreasing as k → ∞. Since µ (B (0, r)) µ k (V k ) − ε∞ 2 k+1 = ∑µ (V k ) − ε 2k=1k=1k=1= µ (V ) − ε 2 ≥ µ (V ) + µ (U) − ε 2 − ε ≥ µ (V ∪ U) − ε2which shows µ is outer regular. Inner regularity can be obtained from Lemma 7.4.3.Alternatively, you can use the above construction to get it right away. It is easier thanthe outer regularity.First assume µ (F ) < ∞. By the first part, there exists a compact set,K k ⊆ F ∩ (B (0, k) \ D (0, k − 1))