Finding a Seq. of Complex Radon Measures on R that Tends to 0 Vaguely

In summary, we are looking for a sequence (mn) of complex Radon measures on R that tends to 0 vaguely. We can do this by considering delta function type measures, where m_n is delta(0) minus delta(1/n). This sequence satisfies the condition for vague convergence, but for the bounded measurable function g with compact support, we can choose g(x)=1 at x=0 and 0 otherwise, which results in the integral of g being 1 for any m. Therefore, this example satisfies the given criteria.
  • #1
AKG
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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.
 
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  • #2
How about thinking of delta function type measures? Take m_n to be delta(0)-delta(1/n). Then it goes to zero for any continuous function, but if we take g(x)=1 at x=0, 0 otherwise then the integral of g is 1 for any m. Does that work?
 
  • #3
Yes, very nice! Thanks.
 

1. What is a complex Radon measure on R?

A complex Radon measure on R is a mathematical object that assigns a complex number to each measurable set on the real line R. It is used to generalize the concept of Lebesgue measure to more general spaces.

2. What does it mean for a sequence of complex Radon measures on R to tend to 0 vaguely?

A sequence of complex Radon measures on R tends to 0 vaguely if the measures of any bounded set approach 0 as the index of the sequence goes to infinity. This means that the measures become increasingly small as the sequence progresses.

3. Why is finding a sequence of complex Radon measures on R that tends to 0 vaguely important?

Finding such a sequence is important in the study of integration theory and functional analysis. It allows for a more general approach to integration and can be used to prove important theorems and properties in these fields.

4. How is the concept of a sequence of complex Radon measures on R related to the concept of a sequence of functions?

A sequence of complex Radon measures on R can be seen as a generalization of a sequence of functions. In fact, a sequence of functions can be viewed as a sequence of complex Radon measures on R if the functions are integrable with respect to Lebesgue measure.

5. Are there any real-world applications of complex Radon measures on R?

Yes, complex Radon measures on R have applications in fields such as physics, engineering, and finance. They are used to model and analyze complex systems and phenomena, such as quantum mechanics, signal processing, and financial markets.

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