Vol. 19, No. 2, 1966

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On the transformation of integrals in measure space

Robin Ward Chaney

Vol. 19 (1966), No. 2, 229–242

One objective of this paper is to prove a formula for the transformation of integrals by means of a change of variable in purely measure—theoretic setting. The classical prototype of such formulas is the one in which the change of variable is effected by an (appropriately differentiable) one-to-one transformation from some subset of Euclidean n-space Rn onto some other subset of Rn; the jacobian of the transformation plays a key role here. For the present study the transformation which gives the change of variable is no longer assumed to be one-to-one but it is required to satisfy certain standard conditions relative to the measure spaces at hand.

Some of the results presented in this paper can be summarized informally as follows. Let T be a function from a nonempty set S onto a set X, let {S,M} and {X,N} be measure spaces, and let B be a sub-σ-field of M. These entities are subjected to certain standard requirements. Within this basic setting is proved a formula which takes the form

∫              ∫
(H ∘T)f dμ =    H ′W  (.,B)dν;
B             T B

in (1), H is some N-measurable function, B is a set in B, f is analogous to the jacobian, and W is a function having certain measure-theoretic properties. Indeed W(x,B) is intended to “count or weigh” the number of points in B mapped into x by T. In this paper certain theorems are proved which reveal in detail the relationship between f and W.

Mathematical Subject Classification
Primary: 28.25
Received: 19 August 1965
Published: 1 November 1966
Robin Ward Chaney