CoNRAN — The Riemann Integral and Measurable Sets. 5 



Hence, has a limit as n increases indefinitely. 



Similarly, has a limit, and the two limits are equal. 



This common limit may now be defined to be the upper integral in E. 

 The independence of the L and S sequences proves that the limit is unique. 

 The lower integral is defined in an analogous manner. 

 Cor. — The integrals in a measurable set are numerically less than M. I. 



5. A fundamental property of these integrals is gi\ei\ in the following- 

 theorem : — 



If a measurable set A is the sum of two nieasurahle sets B and C ivhieh have 

 no point eommon, then the 



I , I integral in A = i , ] integral in B + ( , ] integral in G. 



\loimr J "^ \lowerJ ^ \lowcr j "^ 



Fii'st suppose that A, B, and G are closed sets, and let d be the minimum 



distance (ecart) of the sets B and G. Enclose each point of A in an interval 



of length < ^. By the Heine-Borel theorem, a finite number of these intervals 



suffice to cover all the points of A. l^et Zi denote those of the intervals that 

 contain B, and Zs those that contain G. Z, and Z2 do not overlap. Hence, 

 if Bi be the black intervals of ^ in Zi, and Z's the black intervals of G in Zj, 

 it is evident that A + D% constitute the black intervals of A in Zi + Z,. The 

 theorem follows by § 3. 



If the sets are not closed, we need only consider their closed components ; 

 the theorem then follows from the definitions in the preceding paragraph. 



Theorem : — 



6. The upper and lower integrals of a function taken over the ■measurable set 

 eonsisting of its ^Joints of continuity are eqvxtl. 



Lemma. — If the saltus < k at all j)oints in the open interval « < x < b, 

 then _ 



f - I <zk{l-a). 



Por a closed interval this follows from a theorem due to Baire.* The truth 



of the lemma in our case is apparent from the fact that the 1 , 1 integral 



in the open interval is limit of the [■, , j integral in the closed interval 



a + e < X < b - e, 

 when the positive quantity e is indefinitely di mini shed. 



Annali di Mathematica, Ser. iii, vol. iii, p. 15. 1S99. 



