Definition
Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space Then a regular conditional probability is defined as a function called a "transition probability", where is a valid probability measure (in its second argument) on for all and a measurable function in E (in its first argument) for all such that for all and all
To express this in our more familiar notation:
where i.e. the topological support of the pushforward measure 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.
Read more about this topic: Regular Conditional Probability
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