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

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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:

$\map {L^2} {\Omega, \Sigma, \Pr} \subseteq \map {L^1} {\Omega, \Sigma, \Pr}$

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 $G \in \GG$:

\(\ds \int_G \map P X \rd \Pr\) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_G \map P {X_n} \rd \Pr\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_G X_n \rd \Pr\) 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

$\blacksquare$

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