Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra

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

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\mathcal G \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.

Then there exists an integrable random variable $Z$ on $\struct {\Omega, \GG, \Pr}$ such that:


 * $\ds \int_G Z \rd \Pr = \int_G X \rd \Pr$ for each $G \in \mathcal G$.

Further, if $Z$ and $Z'$ are two integrable random variables satisfying this condition, we have:


 * $Z = Z'$ almost everywhere.

Proof
First take $X \ge 0$.

Define a function $\mu : \GG \to \R$ by:


 * $\ds \map \mu A = \int_A X \rd \Pr$

for each $A \in \GG$.

From Integral of Integrable Function over Measurable Set is Well-Defined, this is well-defined.

From Measure with Density is Measure, $\mu$ is a measure.

Note that if $\map \Pr A = 0$ for $A \in \GG$, we have $\map \mu A = 0$ from Integral of Integrable Function over Null Set.

So $\mu$ is absolutely continuous with respect to $\Pr$.