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regular conditional probability∗ CWoo† 2013-03-21 21:30:55 Introduction Suppose (Ω, F, P ) is a probability space and B ∈ F be an event with P (B) > 0. It is easy to see that PB : F → [0, 1] defined by PB (A) := P (A|B), the conditional probability of event A given B, is a probability measure defined on F, since: 1. PB is clearly non-negative; 2. PB (Ω) = P (B) P (Ω ∩ B) = = 1; P (B) P (B) 3. PB is countably additive: for if {A1 , A2 , . . .} is a countable collection of pairwise disjoint events in F, then S S P ∞ ∞ [ P B ∩ ( Ai ) P (B ∩ Ai ) P (B ∩ Ai ) X PB ( A i ) = = = = PB (Ai ), P (B) P (B) P (B) i=1 i=1 as {B ∩ A1 , B ∩ A2 , . . .} 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 F instead of an event? First, we need to be careful what we mean by PG , since, given any event A ∈ F, P (A|G) is not a real number. And strictly speaking, it is not even a random variable, but an equivalence class 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 F and a sub sigma algebra G of F. A function PG : G × Ω → [0, 1] is a called a regular conditional probability if it has the following properties: ∗ hRegularConditionalProbabilityi created: h2013-03-21i by: hCWooi version: h38574i Privacy setting: h1i hDefinitioni h60A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 1. for each event A ∈ G, PG (A, ·) : Ω → [0, 1] is a conditional probability (as a random variable) of A given G; that is, (a) PG (A, ·) is G-measurable and Z (b) for every B ∈ G, we have PG (A, ·)dP = P (A ∩ B). B 2. for every outcome ω ∈ Ω, 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 “conditional” probability measure on Ω for each outcome in the sense of the first two paragraphs. 2