Let . The Dirac measure concentrated at is defined by
Note that the Dirac measure is indeed a measure:
Since , we have .
Also note that is a probability space.
In other words, integration with respect to the Dirac measure amounts to evaluating the function at .
Moreover, if is defined so that for all , the above becomes
In other words, the function (with fixed and a real variable) behaves like a Radon-Nikodym derivative of with respect to .
Note that, just as the Dirac delta function is a misnomer (it is not really a function), there is not really a Radon-Nikodym derivative of with respect to , since is not absolutely continuous with respect to .
|Date of creation||2013-03-22 17:19:40|
|Last modified on||2013-03-22 17:19:40|
|Last modified by||Wkbj79 (1863)|