Section 1

Gauges and Integrals

The symbol JR . always denotes the real number system, the properties of

which we assume to be familiar to the reader. Our principal elementary

reference will be the third edition of the book of the author and D. R. Sher-

bert, which will be referred to as [B-S], but there are many other books at

that level to which the reader can refer.

Our terminology and notation are standard. If A and B are subsets of

a set X, we denote their union by i U B , their intersection by A n B, and

the relative complement by A — B. If X is understood we sometimes denote

X - A by

Ac.

We denote the distance between two real numbers x,y by

dist(#, y) := \x — y\.

It is clear that, for all x, y, z G R, then

(0) dist(x,y) 0 ;

(1) dist(x, y) = 0 if and only if x = y;

(2) dist(x, y) = dist(y, x)\

(3) dist(x, z) dist(x, y) + dist(y, z).

If x G M . and i C l , where A ^ 0, we sometimes write

dist(x, A) : — inf {dist(x, y) : y G ^4}

for the distance between x and A. The closed neighborhood of x with

radius r 0 is the set

B [ x ; r ] : = { y G M : | x - y | r } ,

3

http://dx.doi.org/10.1090/gsm/032/01