Standard Probability Space - Using The Standardness - Regular Conditional Probabilities

Regular Conditional Probabilities

In the discrete setup, the conditional probability is another probability measure, and the conditional expectation may be treated as the (usual) expectation with respect to the conditional measure, see conditional expectation. In the non-discrete setup, conditioning is often treated indirectly, since the condition may have probability 0, see conditional expectation. As a result, a number of well-known facts have special 'conditional' counterparts. For example: linearity of the expectation; Jensen's inequality (see conditional expectation); Hölder's inequality; the monotone convergence theorem, etc.

Given a random variable on a probability space, it is natural to try constructing a conditional measure, that is, the conditional distribution of given . In general this is impossible (see Durrett 1996, Sect. 4.1(c)). However, for a standard probability space this is possible, and well known as canonical system of measures (see Rokhlin 1962, Sect. 3.1), which is basically the same as conditional probability measures (see Itô 1984, Sect. 3.5), disintegration of measure (see Kechris 1995, Exercise (17.35)), and regular conditional probabilities (see Durrett 1996, Sect. 4.1(c)).

The conditional Jensen's inequality is just the (usual) Jensen's inequality applied to the conditional measure. The same holds for many other facts.