Conditional Expectation is Linear

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

Let $X$ and $Y$ be integrable random variables.

Let $\alpha, \beta \in \R$.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Let $\expect {Y \mid \GG}$ be a version of the conditional expectation of $Y$ given $\GG$.

Let $\expect {\alpha X + \beta Y \mid \GG}$ be a version of the conditional expectation of $\alpha x + \beta Y$ given $\GG$.

Then:

$\expect {\alpha X + \beta Y \mid \GG} = \alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ almost everywhere.

Proof

We show that $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a version of the conditional expectation of $\alpha x + \beta Y$ given $\GG$.

From Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable, we have that $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a $\GG$-measurable random variable.

Let $A \in \GG$.

Then:

\(\ds \int_A \paren {\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG} } \rd \Pr\)

\(=\)

\(\ds \alpha \int_A \expect {X \mid \GG} \rd \Pr + \beta \int_A \expect {Y \mid \GG} \rd \Pr\)

Integral of Integrable Function is Additive, Integral of Integrable Function is Homogeneous

\(\ds \)

\(=\)

\(\ds \alpha \int_A X \rd \Pr + \beta \int_A Y \rd \Pr\)

Definition of Conditional Expectation on Sigma-Algebra

\(\ds \)

\(=\)

\(\ds \int_A \paren {\alpha X + \beta Y} \rd \Pr\)

Integral of Integrable Function is Additive, Integral of Integrable Function is Homogeneous

So $\alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ is a version of the conditional expectation of $\alpha x + \beta Y$ given $\GG$, and from Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra we have:

$\expect {\alpha X + \beta Y \mid \GG} = \alpha \expect {X \mid \GG} + \beta \expect {Y \mid \GG}$ almost everywhere.

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list