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Axiomatic Definition of AREA.

We assume there exists a class M of measurable sets in the plane and a set function a, whose domain is M, with the following properties:


1. Nonnegative property.

  For each set S in M, we have a(S) >= 0.


2. Additive property.

  If S and T are in M, then (S union T) and (S intersect T) are in M, and we have a(S union T) = a(S) + a(T) - a(S intersect T).


3. Difference property.

  If S and T are in M with (S is a subset of T), then T-S is in M, and we have a(T-S) = a(T) - a(S).


4. Invariance under congruence.

  If a set S is in M and if T is congruent to S, then T is also in M and we have a(S) = a(T).


5. Choice of scale.

  Every rectangle R is in M. If the edges of R have lengths h and k, then a(R) = hk.


6. Exhaustion property.

  Let Q be a set that can be enclosed between two step regions S and T, so that

  (S is a subset of Q, which is a subset of T). -- (1.1)

  If there is one and only one number c which satisfies the inequalities

  a(S) <= c <= a(T)

  for all step regions S and T satisfying (1.1), then Q is measurable and a(Q) = c.