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# B. Proofs of Some Area Theorems

B.1   Theorem (Addition Theorem.) For any bounded sets and in
 (B.2)

and consequently

Proof: We have

and

Hence by the additivity of area
 (B.3)

and
 (B.4)

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

B.5   Corollary (Subadditivity of area.) Let , and let , , be bounded sets in . Then
 (B.6)

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

B.7   Theorem (Monotonicity of Area.) Let be bounded sets such that . Then .

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

Since , it follows that .

B.8   Theorem (Additivity for almost disjoint sets.) Let be a finite set of bounded sets such that and are almost disjoint whenever . Then
 (B.9)

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.

Next: Prerequisites Up: Math 111 Calculus I Previous: A. Hints and Answers   Index
Ray Mayer 2007-09-07