Why complete measure spaces? [duplicate]

measure-theory

Solution 1:

In the theory of stochastic processes it is nearly always assumed that the filtrations we are working with are the ones satisfying the so called usual conditions, i.e. they are right continuous and their initial (and thus every) sigma-field is complete.

While it is hard to give some good philosophical explanation of that state of affairs, one thing is certain: the usual conditions are really needed from the technical point of view, they are necessary to perform many constructions and prove theorems in the theory of stochastic processes.

The general theory of stochastic integration is developed with respect to filtrations satisfying usual conditions and complete probability spaces.

If you would like to see the stochastic integration in its most general form and many deep theorems requiring the usual conditions assumption about filtrations and the completeness assumption about probability spaces see "Probabilities and potentials" by Dellacherie and Meyer and "Semimartingales" by Metivier.

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