Absolutely Continuous 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 Q is absolutely continuous with respect to \fs4 P if

\fs4 \left( P(F) = 0 \Rightarrow Q(F) = 0 \right)\text{, for all } F \in \mathcal{F}

We denote absolute continuity with \fs4 Q \ll P.

Essentially this means that when an event is impossible according to the probability measure \fs4 P, then it should be impossible according to \fs4 Q.

Absolute continuity is used to define equivalent probability measures and the Radon-Nikodym derivative of probability measures.

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