Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra/Proof 2
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
Observe that:
- $\map {L^2} {\Omega, \GG, \Pr} \subseteq \map {L^2} {\Omega, \Sigma, \Pr}$
is a closed linear space.
Let:
- $P : \map {L^2} {\Omega, \Sigma, \Pr} \to \map {L^2} {\Omega, \GG, \Pr}$
be the orthogonal projection.
Observe that for all $f \in \map {L^2} {\Omega, \Sigma, \Pr}$ and $g \in \map {L^2} {\Omega, \GG, \Pr}$:
\(\ds \int \map P f g \rd \Pr\) | \(=\) | \(\ds \int f g \rd \Pr + \int \paren {\map P f - f} g \rd \Pr\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \int f g \rd \Pr\) | as $\map P f - f \in \map {L^2} {\Omega, \GG, \Pr}^\perp$ |
Let $f \in \map {L^2} {\Omega, \Sigma, \Pr}$.
Let:
- $\ds g := \chi_{\set {P f \ge 0} } - \chi_{\set {P f < 0} }$
so that:
- $\size {\map P f} = \map P f g$
Since $g \in \map {L^2} {\Omega, \GG, \Pr}$, we have:
\(\ds \int \size {\map P f} \rd \Pr\) | \(=\) | \(\ds \int \map P f g\rd \Pr\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int f g \rd \Pr\) | by $(1)$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \int \size f \rd \Pr\) | as $\size g \le 1$ |
On the other hand, by Cauchy inequality and the density of simple functions:
- $\map {L^2} {\Omega, \Sigma, \Pr} \subseteq \map {L^1} {\Omega, \Sigma, \Pr}$
Therefore, we can extend:
- $P : \map {L^1} {\Omega, \Sigma, \Pr} \to \map {L^1} {\Omega, \GG, \Pr}$
so that:
- $\ds \forall f \in \map {L^1} {\Omega, \Sigma, \Pr} : \norm {P f}_{\map {L^1} {\Omega, \Sigma, \Pr} } \le \norm f_{\map {L^1} {\Omega, \GG, \Pr} }$
Let $X \in \map {L^1} {\Omega, \Sigma, \Pr}$.
Let $\sequence {X_n} \subseteq \map {L^2} {\Omega, \Sigma, \Pr}$ such that:
- $\ds \lim_{n \mathop \to \infty} \norm {X_n - X}_{\map {L^1} {\Omega, \Sigma, \Pr} }$
Then for each $G \in \GG$:
\(\ds \int_G \map P X \rd \Pr\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \int_G \map P {X_n} \rd \Pr\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \int_G X_n \rd \Pr\) | by $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_G X \rd \Pr\) |
Hence we can choose $Z = \map P X$.
$\Box$
Now let $Z'$ be another integrable random variable that is $\GG$-measurable with:
- $\ds \int_A X \rd \Pr = \int_A Z' \rd \Pr$
for all $A \in \GG$.
Then:
\(\ds \int \size {Z - Z'} \rd \Pr\) | \(=\) | \(\ds \int \paren {Z - Z'} \paren {\chi_{\set {Z \ge Z'} } - \chi_{\set {Z < Z'} } } \rd \Pr\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\set {Z \ge Z'} } Z \rd \Pr - \int_{\set {Z < Z'} } Z \rd \Pr - \int_{\set {Z \ge Z'} } Z' \rd \Pr + \int_{\set {Z < Z'} } Z' \rd \Pr\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\set {Z \ge Z'} } X \rd \Pr - \int_{\set {Z < Z'} } X \rd \Pr - \int_{\set {Z \ge Z'} } X \rd \Pr + \int_{\set {Z < Z'} } X \rd \Pr\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
By Measurable Function Zero A.E. iff Absolute Value has Zero Integral:
- $Z = Z'$ almost everywhere
$\blacksquare$