Thread: Radon measures
View Single Post
AKG
#1
Mar25-07, 02:20 PM
Sci Advisor
HW Helper
P: 2,586
Problem

Find an example of a sequence (mn) of complex Radon measures on R that 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 the total variation of 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]
. (mn) tends to 0 vaguely iff for every continuous function f from R to C that vanishes at infinity, [itex]\int f\, dm_n \to 0[/itex]
. if (mn) is a sequence of complex radon measures on R and Fn(x) = mn({y in R : y < x}), then if ||mn|| are uniformly bounded and the Fn converge pointwise to 0, then (mn) 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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100