MATHEMATICS: E. H. MOORE 
631 
viz., there exists a sequence {o-m! : ct], 0-2, . . , <rm, . . , of sets o- of 
such a nature that for every n and set a including Un \ F(o-) — a\ 
a{(T^} • ? w:ZD:am„ 9 (r^'''"v =) . [ F ((t) - a | ^ 1^, (5) 
viz., there exists a sequence { am} such that for every positive integer 
n there exists a positive integer Wn (depending on n) such that for every 
set (7 including | F((7) — <^ | S^/n. 
Now let the function F involve a parameter w on a range U and sup- 
pose that F((T, u) converges as to a for every u oi U; the limit is a 
single-valued function, say of in notation: 
L F (<r, = <^ (w) (w), (6) 
cr 
that is, 
:3 :.a{(7„^}5;z:3:3w„„5o-''''"'"««.Z) .|F((r,«/)--^(w)|^lM (7) 
viz., for every u oi M there exists a sequence [(Tum] of sets <7 (depending 
on u) such that for every positive integer n there exists a positive integer 
niun (depending on u and n) such that for every a including o-um^^ 
I F((r, u) — (p {u)\ ^ 1//^. 
The convergence is semiuniform over the range U in case a single 
sequence [am] is effective as the sequence [<Jun} for every u of U, and 
it is miiform in case moreover for every n a single positive integer nin 
is effective as the positive integer niun for every u of U, that is, the 
notations : 
L F (<7, = V? {u) {u; semiunif.); (8) 
1, F {(T, u) = (p (u) (u; unif.), (9) 
have the respective meanings: 
a{(r^}9w. :3:.w:ID:am„„ ? o-'^'''""" . ID ,\F{(T,u)-<p(u)\^l/n] (10) 
'3.{(Tm}^n. :ID :.'3.fn„^ u::D:(t^ ZD . |F (0-,^)-^ (u) \ ^ 1/n. (11) 
If u is a numerically valued single-valued function of u on U, we 
define semiuniformity and uniformity of convergence relative to v as 
scale function over the range U, in notation: as in (8, 9) with the paren- 
theses replaced by (u: semiunif. v (u)), {u\ unif. v (u)) respectively, 
by replacing in the definitions (10, 11) the final 1/;^ by [ v (u) \/n. Thus, 
semiuniformity and imiformity are absolute, i.e., relative to the scale 
function v(u) = 1. 
