630 
MATHEMATICS: E. H. MOORE 
The integration process J, undefined in my earlier theory, is in this theory 
defined. Its definition turns on the definition of Hmit which I wish to 
explain in this note. 
By way of example consider on the range (p = 1, 2, 3, . . . , 
n, . . . ) a numerically valued function a{p) with absolutely 
convergent sum Ja, = a{l) + ol(2) + a{3) + . . . . Jo: is the limit 
in the classical sense, as n increases without bound, of ]nOi, = a(i) 
+ q:(2) + . . -\- a{n). Here the definitions of J„q: and of limit 
are of special reference. However, taking (not the first n but) any 
finite set a say of n elements p: pi < p2 < . . < pn, oi the range "ip"', 
we secure definitions of ]^a, = a(pi) + a{p2) + . . + a(pn)j and 
of limit which are of general reference. 
Indeed, consider at once a general class ^ and a numerically valued 
(possibly many-valued) function F on the class @ of all finite sets o- of 
elements p of the range ^. (In the example cited F(o-) = J<^a). We 
say that the number a is the limit as to a of the function F(o-), or that, 
as to 0", F((7) converges to a, in notation: 
L FW = a, ' (1) 
in case for every positive number e there exists a set ae (depending on 
e) of such a nature that for every set a including ere [F^a) — a \ ^ e, 
in symbols: 
^ : =) : a (T, 9 (T^'"^ . =) . I F ((t) - ^ I ^ e. (2) 
If for a set a including ae F((t) is many-valued the understanding is 
that the final inequality holds for every value of F(o-). (The notation 
e denotes a positive number; the notations n, m used below denote posi- 
tive integers.) 
The of (1) is a single-valued functional operation of the type of a 
definite integral, in that it reduces every function F{(t) of the class 
of all functions convergent as to o- to a number a. Accordingly, the 
class ^ being general, this definition of limit (even apart from its use 
in the theory mentioned) belongs to General Integral Analysis. 
In order to obtain definitions of various modes of convergence in case 
the function F involves a parameter we notice the equivalent forms 
(3, 4, 5) of the definition. 
w : 3 : 3 o-„ 9 0- ^'^^ . Z) . I F (o-) - I ^ \/n, (3) 
a{(r^} 9 w:3:(r=''"«.Z) .|F(a-)-^7|^lM 
(4) 
