The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.
In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space . This consists of a state space , an event space , which is a sigma-algebra of subsets of , and a probability measure . The measure is defined as a map satisfying countable additivity and normalised as .
A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set of all bounded measurable functions . This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map , satisfying for and monotone convergence: if is a nonnegative sequence increasing to a bounded limit , then tends to .
In the opposite direction, any nonnegative linear map satisfying monotone convergence and defines a probability measure by . This is the unique measure with respect to which expectation agrees with the linear map, . So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of consisting of indicator functions . In practice, it is often more convenient to start with a different subset of . For example, probability measures on can be defined via their Laplace transform, , which represents the expectation on exponential functions . Generalising to complex-valued random variables, probability measures on are often represented by their characteristic function , which is just the expectation of the complex exponentials . In fact, by the monotone class theorem, we can uniquely represent probability measures on by the expectations on any subset which is closed under taking products and generates the sigma-algebra . Continue reading “Algebraic Probability”