Equivalent Probability Measures

Definition

Consider two probability measures \fs4 P and \fs4 Q on a measure space \fs4 (\Omega, \mathcal{F}) (defining two probability spaces). We say that \fs4 P and \fs4 Q are equivalent if the two measures are absolutely continuous with respect to each other. We denote measure equivalence with \fs4 P\sim Q. Thus

\fs4 P\sim Q \Leftrightarrow (P \ll Q \text{ and }Q\ll P)

This means that the two probability measures must agree on impossible events. If an even \fs4 F is impossible according to one measure, then it should be impossible according to the other.

Equivalent measures are used to define the Radon-Nikodym derivative of probability measures.


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