Uniqueness

The existence proof produced a conditional expectation $\E(X \mid \cG)$. A natural follow-up: how many such random variables are there? The answer is essentially one, up to behavior on a $\Pr$-null set.

Proposition

Proposition: Uniqueness (a.s.)

Statement

Let $X$ be integrable on $(\Omega, \cF, \Pr)$ and $\cG \subseteq \cF$ a sub-$\sigma$-field. If $Y$ and $Y'$ are both conditional expectations of $X$ given $\cG$ (each $\cG$-measurable, each satisfying $\int_A Y \, d\Pr = \int_A X \, d\Pr$ for all $A \in \cG$, and the same for $Y'$), then

$$Y \;\stackrel{\text{a.s.}}{=}\; Y'.$$

Proof

Suppose for contradiction that $\Pr(Y \ne Y') > 0$.

The event $\{Y \ne Y'\}$ splits as $\{Y > Y'\} \cup \{Y < Y'\}$, so at least one of the two has positive probability. By symmetry (swap the roles of $Y$ and $Y'$ if needed), assume

$$A \;:=\; \{Y > Y'\}, \qquad \Pr(A) > 0.$$

$A$ is $\cG$-measurable. Both $Y$ and $Y'$ are $\cG$-measurable, so $Y - Y'$ is $\cG$-measurable, and the strict-inequality preimage

$$A \;=\; \{Y - Y' > 0\}$$

is in $\cG$.

Apply condition (2) on $A$. Since $A \in \cG$ and both $Y, Y'$ satisfy the integration identity against $X$,

$$\int_A Y \, d\Pr \;=\; \int_A X \, d\Pr \;=\; \int_A Y' \, d\Pr.$$

Subtracting,

$$\int_A (Y - Y') \, d\Pr \;=\; 0.$$

The contradiction. On the set $A$, $Y - Y' > 0$ by construction. So the integrand $(Y - Y') \mathbb{1}_A$ is strictly positive on the set $A$, which has $\Pr(A) > 0$. Integrating a strictly positive function over a set of positive measure gives a strictly positive value:

$$\int_A (Y - Y') \, d\Pr \;>\; 0.$$

This contradicts the previous identity. Hence $\Pr(Y \ne Y') = 0$, i.e. $Y = Y'$ a.s.

Versions

The proposition above says that any two conditional expectations of the same $X$ given the same $\cG$ agree on $\Omega$ except on a $\Pr$-null set. A specific representative is called a version of $\E(X \mid \cG)$. Different versions differ only on a $\Pr$-null set, so any statement about $\E(X \mid \cG)$ that is invariant under modification on a null set (such as $\E[Y]$, $\Var(Y)$, or any almost-sure inequality) is unambiguous. When writing $\E(X \mid \cG)$ without qualification, we tacitly mean “a version” and the choice does not matter for any almost-sure statement.

Conditioning on a Random Variable

The general definition conditions on a $\sigma$-field. The elementary notion conditions on a random variable. These are reconciled by reducing the second to the first.

Definition: Conditional Expectation Given a Random Variable

Let $(\Omega, \cF, \Pr)$ be a probability space, $X$ a random variable on it, and $Y$ an integrable random variable, $\E|Y| < \infty$. Define

$$\E(Y \mid X) \;:=\; \E\!\big( Y \,\big|\, \sigma(X) \big),$$

where $\sigma(X)$ is the $\sigma$-field generated by $X$.

This recovers the elementary formulas as special cases. When $X$ is discrete with $\Pr(X = x) > 0$ for each $x$ in its support, $\sigma(X)$ is generated by the partition $\{\{X = x\}\}_x$, and the block-average picture applied to this partition recovers

$$\E(Y \mid X)(\omega) \;=\; \sum_x \E(Y \mid X = x) \, \mathbb{1}_{\{X = x\}}(\omega).$$

More generally, since $\E(Y \mid X)$ is $\sigma(X)$-measurable, the Doob-Dynkin lemma guarantees a Borel-measurable $g : \R \to \R$ with $\E(Y \mid X) = g(X)$ a.s. The function $g$ is the regression function of $Y$ on $X$, and $\E(Y \mid X = x) := g(x)$ recovers the elementary “given $X = x$” notation, this time as a function defined up to $\Pr_X$-null sets (where $\Pr_X$ is the distribution of $X$).