Some sets obeying harmonic synthesis
Tóm tắt
LetX be a (not necessarily closed) subspace of the dual spaceB
*
of a separable Banach spaceB. LetX
1
denote the set of all weak
*
limits of sequences inX. DefineX
a
, for every ordinal numbera, by the inductive rule:X
a
= (U
b
<
a
X
b
)
1
.There is always a countable ordinala such thatX
a
is the weak
*
closure ofX; the first sucha is called theorder ofX inB
*
. LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak
*
dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa.
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