Conditional Probability and Expectation

Abstract

The relations between measurements, that is, measurable functions, and events, that is, members of a σ-field, are explored and used to extend the basic ideas of probability to probability conditioned on measurements and events. Such relations are useful for developing properties of certain special functions such as limiting sample averages arising in the study of ergodic properties of information sources. In addition, they are fundamental to the development and interpretation of conditional probability and conditional expectation, that is, probabilities and expectations when we are given partial knowledge about the outcome of an experiment. Although conditional probability and conditional expectation are both common topics in an advanced probability course, the fact that we are living in standard spaces results in additional properties not present in the most general situation. In particular, we find that the conditional probabilities and expectations have more structure that enables them to be interpreted and often treated much like ordinary unconditional probabilities. In technical terms, there are always be regular versions of conditional probability and we will be able to define conditional expectations constructively as expectations with respect to conditional probability measures.