Vol.. 7, 1921 
MATHEMATICS: N. WIENER 
255 
F{f} = a 0 + J* m<Pi(x)dx + J* f(y)<p*(x,y)dxdy + ... 
+ f ». f .../(*» <&i ... + 
./ 0 •/ 0 
and an even wider class of what may be called Stieltjes analyt ic functionals, 
in which the general term 
J... f f(xi) ...f(x n )<p n (x lt ...,x n )dxi, ... dx n 
0 J 0 
is replaced by the Stieltjes integral 5 
r ... r /w ..j(x n )d^ n (x lf ...,x n ). 
J o •/ 0 
In what follows, we shall confine our discussion to Stieltjes analytic func- 
tionals, which we shall call simply analytic. The problem with which we 
are now concerned is the definition of the average of an analytic functional. 
Now, the first property which any average ought to fulfil is that the average 
of the sum of two functionals should equal the sum of their averages. We 
should expect, therefore, that: 
(a) Over a wide range of cases, the average of a series should equal the 
series of the averages of the terms; 
(b) The average of a Stieltjes integral, single or multiple, of a given 
functional with respect to such parameters as it may contain, should be 
equal to the integral of the average; 
(c) A constant multiplied by the average of a functional should equal 
the average of a constant times the functional. 
In accordance with this, we get the following natural definition of the 
average of the analytic functional F. 
A{F\ =A{a 0 +fAx)dHx) +f 1 Jj(x)f(y)dM^y) + ...} 
= a 0 + A^f Q f(x)dUx) | + a{ f(x) f(y)dh(x,y) } + ... 
= a 0 +f Q A {f(x) } dhix) +flf Q A {f(x)f(y) } dM*,y) + »• 
We have already seen how to determine A } as a polynomial 
in the ^'s. Hence whenever the above series converges, we have now 
a way of obtaining a perfectly definite value f or A { F } . It may be noted 
that every term in the above expression in which the sign of integration 
is repeated an odd number of times is identically zero. 
If A IF} is to behave as we should expect it to behave, there are certain 
properties which it must fulfil, at least over a large and important class 
of cases. Among these are the following: 
