Conditional Expectation Conditioned on Event of Non-Zero Probability

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

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

Let $B \in \Sigma$ be an event with:


 * $\map \Pr B > 0$

Let:


 * $\GG = \map \sigma B = \set {\O, B, B^c, \Omega}$

where $\map \sigma B$ is the $\sigma$-algebra generated by $B$.

Let:


 * $\ds \alpha = \frac {\expect {X \cdot 1_B} } {\map \Pr B}$

and:


 * $\ds \beta = \frac {\expect {X \cdot 1_{B^c} } } {\map \Pr {B^c} }$

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

Then:


 * $\ds \expect {X \mid \GG} = \alpha \cdot 1_B + \beta \cdot 1_{B^c}$ almost everywhere.

Proof
We show that:


 * $\ds Z = \alpha \cdot 1_B + \beta \cdot 1_{B^c}$ is a version of the conditional expectation of $X$ given $\GG$.

We will then be done by Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra.

From Characteristic Function Measurable iff Set Measurable, $1_B$ and $1_{B^c}$ are both measurable functions.

From Pointwise Product of Measurable Functions is Measurable and Pointwise Sum of Measurable Functions is Measurable, we therefore have that $Z$ is a random variable.

So we now verify that the expectations take the correct values.

From Integral of Integrable Function over Null Set, we have:


 * $\expect {Z \cdot 1_\O} = 0 = \expect {X \cdot 1_\O}$

We also have:

and:

Finally, we have: