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)f′ is 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.