Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra/Proof 2

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 $g \in \map {L^2} {\Omega, \GG, \Pr}$:

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:

On the other hand, by Cauchy inequality and the density of simple functions:
 * $\map {L^2} {\Omega, \Sigma, \Pr} \subseteq \map {L^1} {\Omega, \Sigma, \Pr}$

is a dense subspace.

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 $G \in \GG$:

Hence we can choose $Z = \map P X$.

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:

By Measurable Function Zero A.E. iff Absolute Value has Zero Integral:
 * $Z = Z'$ almost everywhere