Conditional expectation is the central operator for probability with partial information. It is simultaneously an integral identity, an \(L^2\)-projection, and the algebra behind martingales.
Defining identity
The defining property is equality of integrals over observed events.
Definition. For \(X\in L^1(\Omega,\mathcal F,P)\) and \(\mathcal G\subseteq\mathcal F\), \(\mathbb E[X\mid\mathcal G]\) is a \(\mathcal G\)-measurable integrable random variable \(Y\) such that \[ \int_GY\,\mathrm{d}P=\int_GX\,\mathrm{d}P,\qquad G\in\mathcal G. \]
Existence, uniqueness, and tower property
The operator is characterized by the defining identity.
Theorem. Conditional expectation exists and is unique almost surely. It is linear, positive, contractive on \(L^p\) for \(1\le p\le\infty\), and if \(\mathcal H\subseteq\mathcal G\), then \[ \mathbb E[\mathbb E[X\mid\mathcal G]\mid\mathcal H]=\mathbb E[X\mid\mathcal H]. \]
Proof. For \(X\ge0\), the measure \[ \nu(A)=\int_AX\,\mathrm{d}P,\qquad A\in\mathcal G, \] is absolutely continuous with respect to \(P|_{\mathcal G}\). Radon-Nikodym gives a nonnegative density \(Y\). Apply this to \(X^+,X^-\) for integrable \(X\). If \(Y_1,Y_2\) are versions, then \(G=\{Y_1>Y_2\}\in\mathcal G\) and \[ \int_G(Y_1-Y_2)\,\mathrm{d}P=0, \] forcing \(P(G)=0\); reverse the roles for uniqueness.
Linearity and positivity follow from the defining integrals and uniqueness. Conditional Jensen gives \[ |\mathbb E[X\mid\mathcal G]|^p \le\mathbb E[|X|^p\mid\mathcal G], \] so integration proves \(L^p\)-contractivity. If \(\mathcal H\subset\mathcal G\), the iterated conditional expectation is \(\mathcal H\)-measurable and has the same integrals as \(X\) on every \(H\in\mathcal H\), proving the tower property.
Theorem (conditional Jensen). If \(\phi\) is convex and the variables involved are integrable, then \[ \phi(\mathbb E[X\mid\mathcal G]) \le\mathbb E[\phi(X)\mid\mathcal G] \quad\text{a.s.} \]
Proof. Represent a finite convex function as the supremum of a countable collection of affine minorants with rational coefficients: \[ \phi(x)=\sup_k(a_kx+b_k). \] For each \(k\), \[ a_k\mathbb E[X\mid\mathcal G]+b_k =\mathbb E[a_kX+b_k\mid\mathcal G] \le\mathbb E[\phi(X)\mid\mathcal G]. \] Take the supremum over \(k\).
Pull-out and projection properties
Measurable factors known under \(\mathcal G\) can be pulled out of conditioning.
Proposition. If \(Z\) is bounded and \(\mathcal G\)-measurable, then \[ \mathbb E[ZX\mid\mathcal G]=Z\mathbb E[X\mid\mathcal G]. \] If \(X\in L^2\), then \(\mathbb E[X\mid\mathcal G]\) is the orthogonal projection of \(X\) onto \(L^2(\mathcal G)\).
Proof. For indicators \(Z=1_G\), the pull-out identity follows from the defining integral identity on sets in \(\mathcal G\). Simple approximation and bounded convergence give bounded \(Z\). In \(L^2\), for every \(Y\in L^2(\mathcal G)\), the pull-out identity gives \(\mathbb E[(X-\mathbb E[X\mid\mathcal G])Y]=0\), which is exactly orthogonality to the closed subspace \(L^2(\mathcal G)\).
Proposition (independence and conditioning). If \(X\in L^1\) is independent of \(\mathcal G\), then \[ \mathbb E[X\mid\mathcal G]=\mathbb EX \quad\text{a.s.} \] More generally, if \(X\) and \(Y\) are independent and \(h(X,Y)\) is integrable, then \[ \mathbb E[h(X,Y)\mid Y] =\psi(Y), \qquad \psi(y)=\mathbb E[h(X,y)]. \]
Proof. For \(G\in\mathcal G\), independence gives \[ \mathbb E[X\mathbf1_G]=\mathbb EX\,P(G), \] first for bounded \(X\) and then by truncation. Thus the constant \(\mathbb EX\) satisfies the defining identity. For the second statement, verify \[ \mathbb E[h(X,Y)g(Y)]=\mathbb E[\psi(Y)g(Y)] \] for bounded measurable \(g\) using the product law and Fubini.
Corollary (variance decomposition). For \(X\in L^2\), \[ \operatorname{Var}(X) =\mathbb E[\operatorname{Var}(X\mid\mathcal G)] +\operatorname{Var}(\mathbb E[X\mid\mathcal G]). \]
Proof. Write \[ X-\mathbb EX =\bigl(X-\mathbb E[X\mid\mathcal G]\bigr) +\bigl(\mathbb E[X\mid\mathcal G]-\mathbb EX\bigr). \] The two terms are orthogonal in \(L^2\), because the second is \(\mathcal G\)-measurable. Pythagoras gives the formula.
Remark (Doob-Dynkin lemma). Conditioning on \(Y\) means conditioning on \(\sigma(Y)\). If \(Y\) takes values in a standard Borel space, every \(\sigma(Y)\)-measurable random variable \(Z\) has a representation \(Z=g(Y)\) for some measurable \(g\), unique \(P_Y\)-almost everywhere.