Calculus: Analytic Geometry and Calculus, with Vectors

722 Appendix 2

Figure A.1 shows three spherical shells the inner and outer radii of

which are 1 and 2. A point P lies in one of these shells if its distance r

Green shell Red shell

Figure A.1

5 subsets \ 3 subsets

Yellow shell

% 2 subsets

from the center is such that 1 <_ r S 2. The shells are identical (or
congruent) except that the first one is green, the second one is red, and
the third one is yellow. It has been proved to be possible to separate
the green shell G into five separate and distinct parts or subsets G1j Gs,
G3, G4, G5, to separate the red shell R into three separate and distinct parts
or subsets R1, R2, R3, and to separate the yellow shell Y into two separate
and distinct parts or subsets Yl and Y2 in such a way that
R1 ' G1, R2 - G2, R3 G3, Y1 '" G4, Y2 G6

where the symbol "-" means "is congruent to." If we make the first

three assumptions listed above and use the symbol ISI to denote the

volume of a set S, we obtain

IGI = IG11 + IG21 + IG31 + IG41 + IG51

= IR1I+IR21+IR31+1111+1Y21

= IRI + IYI = IGI + IGI = 21G1.

This implies that IGI = 0 and contradicts the fourth assumption (A4).

Without undertaking to press very far into theories of volumes (these

theories being a part of the more comprehensive theory of additive set

functions in E3). we point out that it is possible to assign numbers (called

volumes) to some of the sets in E3 in such a way that the following state-

ments are true.

(B1) Some sets in E3, including solid spherical shells having inner and

outer radii for which 0 < rl < r2, have positive volumes.

(B2) If S is a set in E3 which has a volume, then each set in E3 which is

congruent to S has a volume which is equal to the volume of S.

(B3) If a set S in E3 is the union of a finite collection of separate and

distinct subsets each of which has a volume, then S has a volume

and the volume of S is the sum of the volumes of the subsets.

The example involving the colored shells proves the following funda-

mental fact. Whenever volumes are assigned to sets in E3 in such a way