Vol. 42, No. 1, 1972

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A remark on Tonelli’s theorem on integration in product spaces

Arunava Mukherjea

Vol. 42 (1972), No. 1, 177–185

This paper is concerned to show a connection between the validity of Tonelli’s theorem on integration in the product of two measure spaces and the semifiniteness of the product measure. The classical Tonelli theorem is usually stated in a sigma-finite setting. It is shown in this paper, among other things, that in a product measure space, where one of the measures is sigma-finite and other one semifinite (not necessarily sigma-finite), Tonelli’s theorem is valid only if the product measure is semifinite and on the other hand, if the product of any two measures is semifinite, then Tonelli’s theorem is valid.

Mathematical Subject Classification 2000
Primary: 28A35
Received: 1 February 1971
Revised: 15 March 1972
Published: 1 July 1972
Arunava Mukherjea