566 
SUB- AND INFRA-UNIFORM CONVERGENCE 
(5 has a measure < o-, small at pleasure. Thus each point of 36 lies 
within a cube. By Borel’s theorem there exists a finite set of 
these cubes 
such that each point of 36 lies within one of them. But corre 
sponding to the T’s, are layers 
such that in each of them 
| e(x, t) | < e. 
Thus / = (f> subuniformly in X = (Tj, T 2 ••• T r ). Let £ be the 
residual set. Obviously £ < cr. Thus the convergence is infra 
uniform. 
2. As a corollary we have : 
Let 
^0*0 = 2/ tl ... ln Oi ••• x m ) 
converge in 21. Let F be limited, and each f L be limited and R-in- 
tegrable in 21. For F to be R-integrable in 21, it is sufficient that F 
converges infra-uniformly in 21. 
If 21 is complete, this condition is necessary. 
553. Infinite Peaks. 1. Let lim/(a: 1 ••• x m t x ••• £„) = fi(x) in 36, 
t finite or infinite. Although f(x, t) is limited in 36 for each t 
near t, and although fi(x) is also limited in 36, we cannot say that 
(1 
| f (x, t) | < some M 
for any x in 36 and any t near t, as is shown by the following 
Example. Let f(x, t) = — = (f>(a;) = 0, as t= co for x in 
36 = (— QO, go). 
It is easy to see that the peak of f becomes infinitely high as 
n = co. 
In fact, for # = —- f— il. Thus the peak is at least as high 
Vi e