6) Since c is inner regular on 6, o is inner regular on C = R(6)
[6, Corollary to Prop. 7, p. 308]. We now show o is regular on C.
This follows immediately from the following proposition [6, p. 306]:
Suppose that the ring C satisfies the following
condition: for every set A E C there exists a set A' C C
such that AC Int(A').
Then a measure m is regular on C if and only if m is
inner regular on C.
We need to show that C satisfies the condition. Let A E C,
n
then A = l A Ai E 6 disjoint. For each i, denote
1=1
Ai = ((sit ),(s',t')]. Then Bi ((si-1,ti-1),(s-+1,t +
belongs to 6: clearly A C Int B for each i; hence
n n n
A = i A C J Int B C Int (J B.) = Int B, denoting
i=1 i=1 i=1
B = B B. E C. Take A' = B.
We have now an extension o of o to C satisfying:
1) o is additive on C
2) o is regular on C
3) o has finite variation on C.
Then, by a standard theorem of measure theory, o can be extended
uniquely to a Borel measure m of finite variation. This measure
clearly coincides with o on 6, so the theorem is proved. I
Remarks.
1) The theorem proved by Radu is for Rn: we have restricted
ourselves to R2 to enhance the clarity of the proof, but Rn presents
no additional difficulties (except with notation!).