Introduction to Calculus and Classical Analysis by Omar Hijab

Author:Omar Hijab
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Since , this implies that

Now apply (4.5.4) with M and A, there, replaced by M 2 and respectively, obtaining

Proceeding in this manner, we obtain

Sending , we obtain

Since

subadditivity implies that

Hence, we obtain the reverse inequality

By choosing A = R 2, we conclude that the method is valid for measurable, hence open sets. This completes stage one of the derivation of the method.

Next we establish the method for interopen sets. A set is interopen if I is the infinite intersection of a sequence of open sets (G n ), . Of course, every open set is interopen. Also every compact rectangle is interopen (Exercise 4.5.5). The key feature of interopen sets is that any set A can be covered by some interopen set I, , having the same area, (Exercise 4.5.7).

Theorem 4.5.6.

If (M n ) is a sequence of measurable sets, then is measurable.

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