TEGRALS PROPER INTEGRALS . 13 
)rime, 
3. In this connection we should note, however, that the converse 
of 2 is not always true, i.e. if (5 is integrable, then 21 has content 
and 2, 2) holds. This is shown by the following : 
tlie :r-axis, we have 
Example. In the unit square we define the points x, y of 21 thus : 
For rational x, o ^ ^ i 
" S 2 ■ 
For irrational a;, i ^ i 
\<y<^ 
Then 
Then (5 = I for every x in 33 Hence 
(1 
/i-l. 
^58 
»oint of 21 and at the 
2Í, let g — 0. Then 
But 21 = 0, 21 = 1. 
10. 1. Let /(aq x^ be limited in the limited field 21 = 423 • (5. 
W>0, p p 7» 
JJ/W- 0- 
fit f /' 7» 7* 
( 2 
7 < fg = % 
Let us first prove 1). Let 21, 33, (S lie in the spaces 9? m , 9I r , 9? s , 
r -f s = m. Then any cubical division T> divides these spaces into 
cubical cells d 0 d{, d'J of volumes d, d’, d" respectively. Ob 
viously d = d'd". D also divides 33 and each £ into unmixed cells 
3', 3". Let M l = Max/ in one of the cells d c , while M' = Max/ 
in the corresponding cell 8”. Then by 2, 4, 
^/<2df"3"<2Tfd" 
since M L , M,! > 0. Hence 
(2 
23' f / < ZdlIMfi" = 2M L d. 
58 58 
Letting the norm of JD converge to zero, we get 1). We get 
2) by similar reasoning or by using 3, 3 and 1).