562 
SUB- AND INFRA-UNIFORM CONVERGENCE 
For the convergence to he sub-uniform in X, it is necessary that for 
each h in 53, and for each e > 0, there exists a t = ß near t, such that 
lim \e(x, i)| >e. 
(1 
For if the convergence is subuniform, there exists for each e 
and y > 0 a finite set of layers ¡On t in K,*(t) such that 
| e (x, 9 | < « » x in . 
Now the point x=h lies in one of these layers, say in 
Then 
| e(x, /3) | < e , for all x in some V*(h'). 
But then 1) holds. 
2. Example. Let 
00 
F(x~) = '2x n (l — x). 
0 
This is the series considered in 140, Ex. 2. 
F converges uniformly in 21 =( — 1, 1), except at x = 1. 
F m (x) = - x m , 
*=i 
Hence F is not subuniformly convergent in 21. 
Integrability 
551. 1. Infra-uniform Convergence. It often happens that 
f (x 1 X m t-^ ••• tf) — (f> (x^ ••• xf) 
subuniformly in X except possibly at certain points (g = \e\ form 
ing a discrete set. To be more specific, let A be a cubical divi 
sion of in which X lies, of norm 8. Let X A denote those cells 
containing points of X, but none of (g. Since (g is discrete, 
X A = X. Suppose now f==(f> subuniformly in any X A ; we shall 
say the convergence is infra-uniform in X. When there are no 
exceptional points, infra-uniform convergence goes over into sub 
uniform convergence.