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.
