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Find an example of a sequence (m_{n}) of complex Radon measures onRthat tends to 0 vaguely such that for some bounded measurable function g with compact support, [itex]\int g\, dm_n \not\to 0[/itex].

Definitions and facts

. complex measures are finite

. Radon measures are only defined on Borel sets

. if m is a Radon measure and E is Borel, m(E) is the infimum of the m(U), where U are open sets containing E

. if m is Radon and U is open, m(U) is the supremum of the m(K), where K are compact sets contained in U

. if m is a complex measure, there exists a positive measure m' and an m'-measurable function f such that [itex]m(E) = \int _E f\, dm'[/itex]. We express this relation by dm = fdm'.

. if m, m', and f are as above, then thetotal variationof m, denoted |m| is defined by the relation d|m| = |f|dm' (this is well-defined, i.e. if m'' is another positive measure, g an m''-measurable function, such that dm = gdm'', then |f|dm' = |g|dm'')

. the norm of a complex measure m,||m||is defined to be [itex]\int d|m|[/itex]

. (m_{n})tends to 0 vaguelyiff for every continuous function f fromRtoCthat vanishes at infinity, [itex]\int f\, dm_n \to 0[/itex]

. if (m_{n}) is a sequence of complex radon measures onRand F_{n}(x) = m_{n}({y inR: y < x}), then if ||m_{n}|| are uniformly bounded and the F_{n}converge pointwise to 0, then (m_{n}) tends to 0 vaguely

. (there are lots theorems in this section, but none that immediately strike me as relevant)

Attempts

It seems to me that if we can find a sequence of measures and a corresponding bounded measurable function with compact support, we can do so with the restriction that this measurable function g have compact support contained in [0,1]. I've tried a few things but they haven't worked.

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# Radon measures

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