Abstract
In [4] we showed that one can tell whether a submeasure on a Boolean algebra has a control measure or is pathological by comparing the Fréchet-Nikodym topology it generates to the universal measure topology of Graves. We then wondered if a submeasure could be decomposed into a part with a control measure and a part which is pathological or zero. This led to the problem of finding a Lebesgue decomposition for a submeasure on an algebra of sets with respect to a Fréchet-Nikodym topology.In [6] Drewnowski proved a Lebesgue decomposition theorem for exhaustive submeasures with respect to “additivities” and a similar theorem for exhaustive Fréchet-Nikodym topologies. He asked if an exhaustive Fréchet-Nikodym topology could be decomposed with respect to another Fréchet-Nikodym topology. In [12] Traynor showed that the answer is “yes”.
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