existence of the conditional expectation

Let (Ω,,) be a probability spaceMathworldPlanetmath and X be a random variableMathworldPlanetmath. For any σ-algebra 𝒢, we show the existence of the conditional expectation 𝔼[X𝒢]. Although it is possible to do this using the Radon-Nikodym theoremMathworldPlanetmath, a different approach is used here which relies on the completeness of the vector spaceMathworldPlanetmath L2. The defining property of the conditional expectation Y=𝔼[X𝒢] is

𝔼[1GY]=𝔼[1GX] (1)

for sets G𝒢. We shall prove the existence of the conditional expectation for all nonnegative random variables and, more generally, whenever 𝔼[|X|𝒢] is almost surely finite.

First, the conditional expectation of every square-integrable random variable exists.

Theorem 1.

Suppose that E[X2]<. Then there is a G-measurable random variable Y satisfying E[Y2]< and equation (1) is satisfied for all GG.


Consider the norm Y2𝔼[Y2]1/2 on the vector space V=L2(Ω,,) of real valued random variables Y satisfying 𝔼[Y2]< (up to almost everywhere equivalence). This is given by the following inner productMathworldPlanetmath


As Lp-spaces are completePlanetmathPlanetmath, this makes V into a Hilbert spaceMathworldPlanetmath (see also, L2-spaces are Hilbert spaces (http://planetmath.org/L2SpacesAreHilbertSpaces)). Then, UL2(Ω,𝒢,) is a complete, and hence closed, subspacePlanetmathPlanetmath of V.

By the existence of orthogonal projections (http://planetmath.org/ProjectionsAndClosedSubspaces) onto closed subspaces of Hilbert spaces, there is an orthogonal projection π:VU. In particular, πY-Y,Z=0 for all YV and ZU. Setting Y=πX gives


as required. ∎

We can now prove the existence of conditional expectations of nonnegative random variables. Note that here there are no integrability conditions on X.

Theorem 2.

Let X be a nonnegative random variable taking values in R{}. Then, there exists a nonnegative G-measurable random variable Y taking values in R{} and satisfying (1) for all GG. Furthermore, Y is uniquely defined P-almost everywhere (http://planetmath.org/AlmostSurely).


First, let Xn=min(n,X). As this is boundedPlanetmathPlanetmath, theorem 1 says that the conditional expectations Yn=𝔼[Yn𝒢] exist. Furthermore, as X0=0, we may take Y0=0. For any n, setting G={Yn+1<Yn}𝒢 gives


So 1G(Yn-Yn+1) is a nonnegative random variable with nonpositive expectation, hence is almost surely equal to zero. Therefore, Yn+1Yn (almost surely) and, by replacing Yn with the maximum of Y1,Yn we may suppose that (Yn) is an increasing sequence of random variables. Setting Y=supnYn, the monotone convergence theoremMathworldPlanetmath gives


as required.

Finally, suppose that Y~ is also a nonnegative 𝒢-measurable random variable satisfying (1). For any x, setting G={Y~>Y,x>Y} then 1GY is bounded and,


showing that (G)=0. Letting x increase to infinity gives Y~Y (almost surely) and, similarly, YY~ so that Y=Y~ almost surely. ∎

Finally, we show existence of the conditional expectation of every random variable X satisfying 𝔼[|X|𝒢]< almost surely. Note, in particular, that this is satisfied whenever X is integrable, as

Theorem 3.

Let X be a random variable such that E[|X|G]< almost surely. Then, there exists a G-measurable random variable Y such that E[1G|Y|]< and (1) is satisfied for every GG with E[1G|X|]<.

Furthermore, Y is uniquely defined up to P-a.e. equivalence.


The positive and negative parts X+,X- of X satisfy


almost surely. We can therefore set Y±𝔼[X±𝒢] and Y=Y+-Y-.

If G𝒢 satisfies 𝔼[1G|X|]< then 𝔼[1GY±]=𝔼[1GX±]<, so 𝔼[1G|Y|]< and,


as required.

Finally, suppose that Y~ satisfies the same conditions as Y. For any x0 set G={Y++Y-x,Y~>Y}𝒢. Then,


So, 𝔼[1G|Y|] and 𝔼[1G|Y~|] are finite, hence (1) gives


So (G)=0 and, letting x increase to infinity, Y~Y almost surely. Similarly, YY~ and therefore Y~=Y almost surely. ∎

Title existence of the conditional expectation
Canonical name ExistenceOfTheConditionalExpectation
Date of creation 2013-03-22 18:39:28
Last modified on 2013-03-22 18:39:28
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 60A10