Vol. 25, No. 1, 1968

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A chain rule for the transformation of integrals in measure space

Robin Ward Chaney

Vol. 25 (1968), No. 1, 33–57

This paper deals with the composition of two transformations each of which is being used to effect a transformation of an (abstract) integral by means of a change of variable. The principal result is an abstract version of the “chain rule” in a purely measure-theoretic setting. This principal result is another in the line of extensions and variations of the theorem which asserts that if f and g are absolutely continuous real valued functions on suitable closed intervals on the line then g f is absolutely continuous if and only if (g′∘ f)fis integrable over the domain of f; and if g f is absolutely continuous then (g f) = (g′∘ f)f. This theorem has previously been generalized to functions on n-space. In this paper certain results of a similar type are presented in a general measure-theoretic setting.

Mathematical Subject Classification
Primary: 28.16
Received: 25 March 1966
Revised: 21 April 1967
Published: 1 April 1968
Robin Ward Chaney