Mm 
6 POINT SETS AND PROPER INTEGRALS 
S.IfoO, J.tf-cff; 
hMtifc < 0’ fr-'fc 
For in any cell 
Max • cf = c Max/; Min • cf = c Min/ 
when c > 0 ; while 
Max • cf = c Min/; Min •</=<? Max/ 
when c < 0. 
4. If g is integrable in §1, 
f(/+£)=JV + j g- 
For from 
Max / + Min g < Max (/+/)< Max / + Max g, 
we have 
X /+ X^/>+^X^+X^ 
But g being integrable, 
Hence 2) gives 
X/ + J^ = X (/+ ^ 
(1 
(2 
which is the first half of 1). The other half follows from the 
relation 
Min/+ Min g< Min (/ + /) < Min/+ Max g. 
5. The integrands f g being limited, 
f (/ + ,)<f / + jf # < f(/ + /)• 
iAgt ^21 *A2t 
For in any cell d\ 
Min (/ + ¿0 < Min/ + Max# < Max (/ + /). 
6. Let f = ( 
For 
Then by 2 ai 
or 
4. Let f(xj ■ 
Let us effect 
To prove 1) 
customary nota 
Hence 
Letting 8=0 
which is 1).