B. Proofs of Some Area Theorems

Proof: We have

and

Hence by the additivity of area

and

If we solve equation (B.4) for and use this result in equation (B.3) we get the desired result.

Proof: The proof is by induction. If , then (B.6) says , which is true. Suppose now that is a generic element of , and that (B.6) is true when . Let be bounded sets in . Then

Hence (B.6) is true when , and by induction the formula holds for all

Proof: If then , and in this case
equation (B.4)
becomes

Since , it follows that .

Proof: The proof is by induction on . For , equation (B.9)
says
that
, and this is true. Now suppose
is a family of mutually almost-disjoint sets. Then

and this is a finite union of zero-area sets, and hence is a zero-area set. Hence, by the addition theorem,

i.e.,

The theorem now follows from the induction principle.