TY - JOUR
AU - Hobson, E. W.
AB - 10 DR. E. W. HOBSON [Jan. 14, ON CHANGE OF THE VARIABLE IN A LEBESGUE INTEGRAL By E. W. HOBSON. [Received and Read January 14th, 1909.] THE use of the Lebesgue integral, instead of the Riemann integral, has the advantage of imparting much greater generality to the results of investigations in which it is employed. It is therefore desirable that all the processes of the integral calculus should be extended to cases in which the Lebesgue integral takes the place of the Riemann integral. In the present communication I propose to consider the effect of a change of the independent variable in a Lebesgue integral, and to establish theorems relating to the equality of the integral in its original form with the new form obtained by a change of the independent variable, analogous to the known results relating to the Riemann integral. 1. Let the variable x denote points in a limited interval (a, b), and let the variable y denote corresponding points in the limited interval (a, /3), the relation between the two variables being x = yfs(y), where \fr{y) is a continuous monotone function, with no lines of invariability. We shall suppose that \fs {y) increases from 
TI - On Change of the Variable in a Lebesgue Integral
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s2-8.1.10
DA - 1910-01-01
UR - https://www.deepdyve.com/lp/wiley/on-change-of-the-variable-in-a-lebesgue-integral-E2oT74cN6r
SP - 10
EP - 21
VL - s2-8
IS - 1
DP - DeepDyve
ER -