regular conditional probability


Suppose (Ω,,P) is a probability spaceMathworldPlanetmath and B be an event with P(B)>0. It is easy to see that PB:[0,1] defined by


the conditional probabilityMathworldPlanetmath of event A given B, is a probability measure defined on , since:

  1. 1.

    PB is clearly non-negative;

  2. 2.


  3. 3.

    PB is countably additivePlanetmathPlanetmath: for if {A1,A2,} is a countableMathworldPlanetmath collectionMathworldPlanetmath of pairwise disjoint events in , then


    as {BA1,BA2,} is a collection of pairwise disjoint events also.

Regular Conditional Probability

Can we extend the definition above to P𝒢, where 𝒢 is a sub sigma algebra of instead of an event? First, we need to be careful what we mean by P𝒢, since, given any event A, P(A|𝒢) is not a real number. And strictly speaking, it is not even a random variableMathworldPlanetmath, but an equivalence classMathworldPlanetmathPlanetmath of random variables (each pair differing by a null event in 𝒢).

With this in mind, we start with a probability measure P defined on and a sub sigma algebra 𝒢 of . A function P𝒢:𝒢×Ω[0,1] is a called a regular conditional probability if it has the following properties:

  1. 1.

    for each event A𝒢, P𝒢(A,):Ω[0,1] is a conditional probability ( (as a random variable) of A given 𝒢; that is,

    1. (a)

      P𝒢(A,) is 𝒢-measurable ( and

    2. (b)

      for every B𝒢, we have BP𝒢(A,)𝑑P=P(AB).

  2. 2.

    for every outcome ωΩ, P𝒢(,ω):𝒢[0,1] is a probability measure.

There are probability spaces where no regular conditional probabilities can be defined. However, when a regular conditional probability function does exist on a space Ω, then by condition 2 of the definition, we can define a “conditionalMathworldPlanetmathPlanetmath” probability measure on Ω for each outcome in the sense of the first two paragraphs.

Title regular conditional probability
Canonical name RegularConditionalProbability
Date of creation 2013-03-22 16:25:24
Last modified on 2013-03-22 16:25:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 60A99