H6 



DR J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



of intervals* whose number does not exceed M. Let I,,, be the greatest of the values 

 of J | /(<) | dt in the various intervals of j c .,. Clearly we have 



If Lg(s,t,n)-g(s,t)]f(t)dt 



I *jki 



where </ is the upper limit of | g (s, t) \ in the rectangle R Hence, if J,. is portion 

 of the interval a^t^c which is not covered by one or other of the intervals;,,.. 



,, (2) 



| \'g (, , )/() *- j V (', 0/0 rf < ^ | jj? (' '> w )^ (' * )]/W * 



Now a Lebesgue integral is-a continuous function of its upper limit; hence, if e is 

 an arbitrarily assigned positive number, we can choose 77 so small that the Lebesgue 

 integral of | f(t)\ in any interval of length 2r), which lies in (a, b), is less than 



With this choice of y we have 



for all values of c in (a, b) and of s in (a,, fcj), since I c ,, is the value of 



\\f(t)\dt , 



in an interval of length not greater than 2r). 



Again, in virtue of our hypothesis as to the uniform convergence of g (s, t, n), we 

 can find a number N great enough to ensure that 



\g( S ,t,n)-g(s,t)\<e/2\\f(t)\dt 



for all values of s and t satisfying (1), and for all values of n > N. Since the points 

 of J e . satisfy (1) for each pair of values of s and c, we have 



If for(,,n)-0M]/()<fc <[ l 



I JJ'.i .'},., 



which is 



for all values of c in (a, b), of .s in (re t , 6,), and of n > N. It follows from (2) that, for 

 these values of c, s, and n, 



which establishes the general theorem. 



* These intervals may, of course, overlap. 



