Normally we define the conditional probability of an event A given an event B as: The difficulty with this arises when the event B is too small to have a non-zero probability. For example, suppose we have a random variable X with a uniform distribution on and B is the event that Clearly, the probability of B, in this case, is but nonetheless we would still like to assign meaning to a conditional probability such as To do so rigorously requires the definition of a regular conditional probability.
Definition
Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space. One should think of as a way to "disintegrate" the sample space into. Using the disintegration theorem from the measure theory, it allows us to "disintegrate" the measure into a collection of measures, one for each. Formally, a regular conditional probability is defined as a function called a "transition probability", where:
For every, is a probability measure on. Thus we provide one measure for each.
where is the pushforward measure of the distribution of the random element, i.e. the topological support of the. Specifically, if we take, then, and so where can be denoted, using more familiar terms . As can be seen from the integral above, the value of for points x outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of T. The measurable space is said to have the regular conditional probability property if for all probability measures on all random variables on admit a regular conditional probability. A Radon space, in particular, has this property. See also conditional probability and conditional probability distribution.
Alternate definition
Consider a Radonspace and a real-valued random variable T. As discussed above, in this case there exists a regular conditional probability with respect toT. Moreover, we can alternatively define the regular conditional probability for an event A given a particular value t of the random variable T in the following manner: where the limit is taken over the net of openneighborhoodsU of t as they become smaller with respect to set inclusion. This limit is defined if and only if the probability space is Radon, and only in the support of T, as described in the article. This is the restriction of the transition probability to the support of T. To describe this limiting process rigorously: For every there exists an open neighborhoodU of the event, such that for every open V with where is the limit.
Example
To continue with our motivating example above, we consider a real-valued random variable X and write This limit, if it exists, is a regular conditional probability for X, restricted to In any case, it is easy to see that this limit fails to exist for outside the support of X: since the support of a random variable is defined as the set of all points in its state space whose every neighborhood has positive probability, for every point outside the support of X there will be an such that Thus if X is distributed uniformly on it is truly meaningless to condition a probability on "".
