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!).