142 
of «. Now let the exterior measure of D be w times the volume 
of the beam H. If 4 describes the sequence of points of which the 
coordinates are powers of 2 with integer non-negative exponents, 
H \ 
the beams — occupy together exactly the set of points B. The set 
‘ ui 
— lies in — at a positive distance from the boundary and the 
q q 
Hb ee leg 
exterior measure of — is w times the volume of —. The exterior 
Ly 
HL ge 
measure of the set formed by all — is therefore wb. Any measur- 
1 
A D . es 
able set containing all the sets — has accordingly at the origin a 
1 
. D 
metric density 2 w. In each of the sets —, p (£) contains at least one 
| 
term which is infinite, so that p (§) is there infinite or indeterminate; 
any measurable set where g(§) has a definite finite value, has (here- 
fore at the origin a metric density <1—w, so that f(&) is not 
integrable (C). 
Proposition 5: If f, (§) and f,(&) are integrable (C) and e, ande, 
represent two arbitrary finite numbers, a function coinciding with 
ef) +e, f, (8) when this expression has meaning, is integrable 
(C) and the integral is e, times the integral of f, (8), augmented by 
e‚ times the integral of f, (8). 
Proof: Two functions are called equivalent if they coincide except 
perhaps in a set with measure zero. It appears from the preceding 
proposition that any function which is integrable (C’), is equivalent 
to a finite function, from the first proposition that two equivalent 
functions are either both or neither integrable (C). As the above 
mentioned proposition for finite functions follows immediately from 
the definition of the integral notion (C), the property holds generally, 
because, if necessary, the functions can first be replaced by the 
equivalent finite functions. 
Proposition 6: For a monotone series of measurable (C) functions 
HE (t= 1, 2,...) approaching to f(&), the series of the integrals is 
likewise monotone. Further the limit function is integrable (C) only 
when the series of the integrals is limited. If this is the case, the 
integral of f(§) is equal to the limit for t= of the integral of 
file): 
Proof: If f(§) is integrable (C), the integral is not less, resp. 
not more, than that of 7,(§), so that the non-decreasing, resp. non- 
