8.7. REGULAR MEASURES 191

If∞

∑k=1

µ(S∩ (Kk \Kk+1))< ∞, (8.14)

then µ(S∩ (Kn \K))→ 0 because it is dominated by the tail of a convergent series so itsuffices to show 8.14.

M

∑k=1

µ(S∩ (Kk \Kk+1)) =

∑k even, k≤M

µ(S∩ (Kk \Kk+1))+ ∑k odd, k≤M

µ(S∩ (Kk \Kk+1)). (8.15)

By the construction, the distance between any pair of sets, S∩(Kk \Kk+1) for different evenvalues of k is positive and the distance between any pair of sets, S∩(Kk \Kk+1) for differentodd values of k is positive. Therefore,

∑k even, k≤M

µ(S∩ (Kk \Kk+1))+ ∑k odd, k≤M

µ(S∩ (Kk \Kk+1))≤

µ

( ⋃k even, k≤M

(S∩ (Kk \Kk+1))

)+µ

( ⋃k odd, k≤M

(S∩ (Kk \Kk+1))

)≤ µ (S)+µ (S) = 2µ (S)

and so for all M, ∑Mk=1 µ(S∩ (Kk \Kk+1))≤ 2µ (S) showing 8.14. ■

8.7 Regular MeasuresIn using measures defined on a σ algebra of subsets of a metric space, the idea of regularityis fundamental.

Definition 8.7.1 A measure µ defined on a σ algebra F of sets in a metric spaceX which includes the Borel sets B (X) will be called inner regular on F if for all F ∈F ,

µ (F) = sup{µ (K) : K ⊆ F and K is closed} (8.16)

A measure, µ defined on F will be called outer regular on F if for all F ∈F ,

µ (F) = inf{µ (V ) : V ⊇ F and V is open} (8.17)

When a measure is both inner and outer regular, it is called regular. Actually, it is moreuseful and likely more standard to refer to µ being inner regular as

µ (F) = sup{µ (K) : K ⊆ F and K is compact} (8.18)

Thus the word “closed” is replaced with “compact”. A complete measure defined on a σ

algebra F which includes the Borel sets which is finite on compact sets and also satisfies8.17 and 8.18 for each F ∈F is called a Radon measure. A Gδ set, pronounced as G deltais the countable intersection of open sets. An Fσ set, pronounced F sigma is the countableunion of closed sets.

In every case which has been of interest to me, the measure has been σ finite.