236 Integrals
escape this awkward situation with the aid of a definition designed for
the purpose.
Definition 4.42 If R is a rectangular region having base length b and
height h, then the product bh is called the area of R. This area is denotedt
by the symbol IRI so that JRI = bh.
This takes care of the matter of areas of rectangular regions, but we
are not yet out of trouble. Let T be the triangular set consisting of those
points within the rectangular region of Figure 4.41 which lie on and
beneath the diagonal drawn there. When we try to decide whether there
is a number which is the area of T, we find that we still need definitions
or postulates or something before we can do anything. If we try to
take care of triangular regions, circular disks, circular sectors, and sets
of other special types by hordes of special definitions, we will find our-
selves forever wallowing in confusion. While students in elementary
calculus courses are normally not expected to know much if anything
about the matter, we should at least know that our friend Lebesgue con-
structed a theory of area which is usually called the theory of Lebesgue
Figure 4.43two-dimensional measure. This eliminates the
confusion and is now very important in applied
as well as in pure mathematics. We should not
be injured and may possibly be benefited by a brief
look at the Lebesgue theory. Let S be a set of
points (x,y) which is contained in but does not
completely fill a rectangle R. Figure 4.43 may be
helpful, but may also be misleading because the set S need not look
at all like the one shown in the figure. Let S' be the set of points in R
but not in S.
Definition 4.44 The set S is said to have area (or two-dimensional
Lebesgue measure) ISI if ISI is a number such that to each e > 0 there corre-
spond (i) a countable collection R1, R2, R3, of rectangular regions such
that each point of S lies in at least one of these regions and(4.45) 1R51 + JR2I + - + IR.1 < IS! + e (n = 1, 2, 3, ...)
and (ii) another countable collection Ri, R2, R3, - of rectangular regions
such that each point of S' lies in at least one of these regions and(4.46) IRil + IRJJ + ... + IRJJ < IRI - BSI + e (n = 1,2,3, ...).
f This notation accords with a general principle with which we are slowly becoming
acquainted. If Q is a number or a partition or a point set or perhaps even an assertion or a
crate of oranges, we expect IQI to be a real nonnegative number which is associated with Q
in some particular way and is, in some sense or other, a measure or a norm or a value of Q.
The simplest useful example is that in which Q is a real number and IQI is its absolute value.
When applications of areas are involved, it is often necessary to recognize that h and k are
numbers representing lengths measured in particular units (say centimeters) and that the
area is a number of appropriate "square units" (say, square centimeters).