I wonder about motivations of a work of Ionescu Tulcea-Marinescu. In order to establish the decomposition of the operator $T$ they assume (condition (1.3)) this operator satisfies the inequality
$$\|Tx\| \leq r.\|x\|+R.|x|$$
We find this type of inequality in the work of Lasota-Yorke as well. I wonder about the motivation of this inequality for Ionescu Tulcea-Marinescu.