Regular Conditional Probability - Alternate Definition

Alternate Definition

Consider a Radon space (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable T. As discussed above, in this case there exists a regular conditional probability with respect to T. 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 open neighborhoods U 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 neighborhood U of t, such that for every open V with

where is the limit.