60 Transactions of the Academy of Science of St. Louts 
measurable, then an /-function is measurable. 
In the majority of cases the proofs of theorems which we 
shall now begin to state are omitted because they are to be Car- 
ried through in exactly the same way as the corresponding theo- 
rems in ordinary analysis. In other cases we shall assume that 
the proof is of sufficient novelty or interest to be given explicity. 
Also, we shall state a theorem with great care only when it is 
considered to be of sufficient importance to do so. 
A. The scalar multiple of an M-function (or continuous 
function) is an M-function (or continuous function). 
T.2. The sum of two M-functions (continuous functions) is 
an \/-function (continuous function). 
T.3. The product of two M-functions (continuous func- 
tions), providing the product has meaning, is an /-function 
‘(continuous function). 
n M-function (continuous function) of an M-function 
(continuous function) is an M-function (continuous function). 
T5. if f,(~) on S,(S, 6 E,,) to S’(S'¢ Z,,) is an M-fuse 
tion, and if {f,(x) } is convergent to f(x) on T (T ¢ II*S,), where 
II”S,, indicates the “durchschnitt” of the sequence, then f(x) is 
an \M-function on 7 to S’. 
Proof: From the fact that {f,(x)} is convergent it follows 
that there exists a sequence of sets iT} with 7 as its outer 
limiting set, and such that {f,(x)} is uniformly convergent on 
each 7,,,. To prove this let {pcx } be a sequence of positive num- 
bers tending to zero, and {nx(px, x) } the corresponding sequence 
of integers associated with {f,(x)}, such that ||f,(x) —f(x)|| <ps 
whenever »>mx(px, x). Let 7; be the set of points for whic 
n.(px, x) is bounded as x travels over J, and when m;(px) is the 
least upper bound. The sequence {mr(px) } then satisfies the 
condition that {f,(x)} be uniformly convergent on 7;. By 
steps, increasing the value of the upper bound, we obtain the 
desired sequence {7,,}. The theorem is now obvious on tak- 
ing into account the fact that a uniformly convergent sequence 
of continuous functions converges to a continuous function. 
Nullifying elements. We say that a,,) is a nullifying element 
if a(,.x=0 for every x in Eq). However it is quite sufficient 
for a@;,)x to vanish in some neighborhood of x = x9, say whenever 
\|x—x0l| <p. For if this be true, whatever y may be, diayy =9, 
because we may write QA (ny VY = Ain) (ay+x9), and we may choose 
a so that || exy| <p. 
If the norm of @,,) is so defined that in the inequality 
