Probability Space

Definition

A probability space \fs4 (\Omega, \mathscr{F}, P) is a measure space with total measure one (that is to say \fs4 P(\Omega)=1). The triplet \fs4 (\Omega, \mathscr{F}, P) is defined as follows:

  • The set \fs4 \Omega is the state space, and its elements \fs4 \omega \in \Omega are typically called states of nature or outcomes. If we think that we are carrying out an experiment, then the set \fs4 \Omega will be the set of all possible outcomes.
  • The set \fs4 \mathscr{F} is the set of events, which consists of some (but not all) subsets of \fs4 \Omega: if \fs4 F \in \mathscr{F} then \fs4 F \subseteq \Omega. Intuitively, the events set is the set of outcomes that want to attach probabilities to. Not all subsets of the state space are valid events, in particular \fs4 \mathscr{F} must be a \fs4 \sigma-algebra on \fs4 \Omega.
  • The function \fs4 P: \mathscr{F} \rightarrow [0, 1] is a probability measure (that is a measure with \fs4 P(\Omega)=1).


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