The first three sections of this appendix are concerned with notation and terminology. Readers should particularly note the standing assumptions such as joint measurability of stochastic processes. The last two sections are brief, stating without proof a basic result from martingale theory and a useful version of Fubini's theorem.

A filtered probability space

In the mathematical theory of probability, one begins with an abstract space Ω, a σ-algebra ℱ on Ω, and a probability measure P on (Ω, ℱ). The pair (Ω, ℱ) is called a measurable space and the triple (Ω, ℱ, P) is called a probability space. Individual points ω ∈ Ω represent possible outcomes for some experiment (broadly defined) in which we are interested. Identifying an appropriate outcome space Ω is always the first step in probabilistic modeling. Then F specifies the set of all events(subsets of Ω) to which we are prepared to assign probability numbers. Finally, the probability numbers P(·) reflect the relative likelihood of various events, whatever that may be interpreted to mean, and their specification is the second major step in probabilistic modeling. In economics one frequently interprets the probability measure P as a quantification of the subjective uncertainty experienced by some rational economic agent. Most physical scientists feel the need for a stronger, objective interpretation related to physical frequency.