Weak limit of abs. continuous measures

Thread starter tom85
Start date Sep 23, 2011
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Continuous Limit Weak

In summary, a weak limit of absolutely continuous measures is a form of convergence in measure theory that only requires convergence of integrals over test functions. This is different from the strong limit, which requires pointwise convergence. The weak limit has applications in probability theory, mathematical physics, and other areas of mathematics. It is also related to the Radon-Nikodym derivative, as it can be used to determine if two measures are absolutely continuous with respect to each other. The weak limit can also be used to prove theorems in measure theory, particularly in the study of convergence of random variables and stochastic processes.

Sep 23, 2011

tom85

say we have a sequence of probability measures on R, such that each one is abs. continuous wrt the Lebesgue measure...

is it possible that these measures converge weakly to a measure which is _not_ abs. continuous wrt the Lebesgue measure?

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Sep 26, 2011

bpet

yes, e.g. normal distribution as variance goes to zero.

Related to Weak limit of abs. continuous measures

1. What is a weak limit of absolutely continuous measures?

A weak limit of absolutely continuous measures is a concept in measure theory that describes the convergence of a sequence of measures to a limit measure. It is called "weak" because it only requires convergence of integrals over certain test functions, rather than pointwise convergence.

2. How is the weak limit of absolutely continuous measures different from the strong limit?

The strong limit of a sequence of measures requires pointwise convergence, meaning that the measures of all sets must converge to the corresponding sets in the limit. The weak limit, on the other hand, only requires convergence of integrals over certain test functions. This means that the weak limit is a weaker form of convergence, but it is still useful in certain contexts.

3. What are some applications of the weak limit of absolutely continuous measures?

The weak limit of absolutely continuous measures has many applications in probability theory, particularly in the study of stochastic processes. It is also used in mathematical physics, statistics, and other areas of mathematics.

4. How is the weak limit of absolutely continuous measures related to the Radon-Nikodym derivative?

The Radon-Nikodym derivative is a tool used to describe the relationship between two measures. The weak limit of absolutely continuous measures is related to this derivative because it can be used to determine if two measures are absolutely continuous with respect to each other.

5. Can the weak limit of absolutely continuous measures be used to prove theorems?

Yes, the weak limit of absolutely continuous measures is an important tool in measure theory and can be used to prove many theorems. It is particularly useful in the study of convergence of random variables and in proving theorems related to stochastic processes and probability theory.