564 
SUB- AND INFRA-UNIFORM CONVERGENCE 
It is sufficient. We show that for each e, <w > 0 there exists a 
division D of such that the cells in which 
Osc (/>>&> 
(1 
have a volume < <r. For setting as usual 
/=<£+e, 
we have in any point set, 
Osc cf> < Osc/+ Osc e. 
Using the notation of 551, 
!<*, 01 <5 
in the finite set of deleted layers i x , i 2 ••• corresponding to 
t=t 1 , For each of these ordinates t t ,f(x, i t ) is integrable 
in X. There exists, therefore, a rectangular division D of 9? m , 
such that those cells in which 
division of such that the cells containing points of the residual 
set £ have a content < <x/2. Let F = D + Then those cells 
of F in which 
Osc f(x, O >", or Ose I e(x, it) I >| 
¿ = 1, 2 ••• have a content < a. Hence those cells in which 1) 
holds have a content < <r. 
It is necessary, if£ is complete. For let 
T. 
Since (f> and f(x, i n ) are integrable, the points of discontinuity of 
</>(#) and of f(x, i n ) are null sets by 462, 6. Hence if (£, denote 
the points of continuity of <f)(x) and f (x, t) in X, 
<£ = <£, = £ 
since X is measurable, as it is complete.