Regular conditional probability

Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. It is defined as an alternative probability measure conditioned on a particular value of a random variable. Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. It is defined as an alternative probability measure conditioned on a particular value of a random variable. 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 [ 0 , 1 ] , {displaystyle ,} and B is the event that X = 2 / 3. {displaystyle X=2/3.} Clearly, the probability of B, in this case, is P ( B ) = 0 , {displaystyle P(B)=0,} but nonetheless we would still like to assign meaning to a conditional probability such as P ( A | X = 2 / 3 ) . {displaystyle P(A|X=2/3).} To do so rigorously requires the definition of a regular conditional probability. Let ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} be a probability space, and let T : Ω → E {displaystyle T:Omega ightarrow E} be a random variable, defined as a Borel-measurable function from Ω {displaystyle Omega } to its state space ( E , E ) . {displaystyle (E,{mathcal {E}}).} Then a regular conditional probability is defined as a function ν : E × F → [ 0 , 1 ] , {displaystyle u :E imes {mathcal {F}} ightarrow ,} called a 'transition probability', where ν ( x , A ) {displaystyle u (x,A)} is a valid probability measure (in its second argument) on F {displaystyle {mathcal {F}}} for all x ∈ E {displaystyle xin E} and a measurable function in E (in its first argument) for all A ∈ F , {displaystyle Ain {mathcal {F}},} such that for all A ∈ F {displaystyle Ain {mathcal {F}}} and all B ∈ E {displaystyle Bin {mathcal {E}}}