Four counterexamples to the Fubini theorem

In this paper we study signed measures. Our main results are as follows: the Fubini theorem is not true in the general case; the Jordan parts of a transition measure are not necessarily transition measures; the operation of taking the Jordan parts does not necessarily commute with multiplying by the initial measure; the product of σ-bounded measures need not be a σ-bounded measure. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 124–127, July, 1997. Translated by S. S. Anisov