FUNCTION THEORY OF REAL 
VARIABLES 
CHAPTER I 
POINT SETS AND PROPER INTEGRALS 
1. In this short chapter we wish to complete our treatment of 
proper multiple integrals and give a few theorems on point sets 
which we shall either need now or in the next chapter where we 
take up the important subject of improper multiple integrals. 
In Volume I, 702, we have said that a limited point set whose 
upper and lower contents are the same is measurable. It seems 
best to reserve this term for another notion which has come into 
great prominence of late. We shall therefore in the future call 
sets whose upper and lower contents are equal, metric sets. When 
a set 21 is metric, either symbol 
I or 21 
expresses its content. In the following it will be often con 
venient to denote the content of 21 by 
21. 
This notation will serve to keep in mind that 21 is metric, when 
we are reasoning with sets some of which are metric, and some 
are not. 
The frontier of a set as 21, may be denoted by 
Front 21. 
2. 1. In I, 713 we have introduced the very general notion of 
cell, division of space into cells, etc. The definition as there 
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