We show that there is a compact topological space carrying a measure which is
not a weak* limit of finitely supported measures but is in the sequential
closure of the set of such measures. We construct compact spaces with measures
of arbitrarily high levels of complexity in this sequential hierarchy. It
follows that there is a compact space in which the sequential closure cannot be
obtained in countably many steps. However, we show that this is not the case
for our spaces where the sequential closure is always obtained in countably
many steps.Comment: (18 pages, a gap in an argument from the previous version fixed
We investigate Talagrand’s construction of an exhaustive pathological submeasure. We consider the forcing associated with this submeasure and we also begin an effort to explicitly describe this construction.
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