Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space, $\mathbb K \in \{\mathbb R, \mathbb C\}$, and $i$ the imaginary unit. Below is the Corollary 2.12 from Amann's Analysis III.
Corollary 2.12
- (i) $f=(f_1, \ldots, f_n) \in L_1(X, \mathcal F, \mu, \mathbb K^n)$ if and only $f_j \in L_1(X, \mathcal F, \mu, \mathbb K)$ for all $j$. In this case, $$ \int_X f \mathrm d \mu = \left ( \int_X f_1 \mathrm d \mu, \ldots, \int_X f_n \mathrm d \mu \right ). $$
- (ii) Suppose $g, h:X \to \mathbb R$ and define $f:=g+ih$. Then $f\in L_1(X, \mathcal F, \mu, \mathbb C)$ if and only if $g,h \in L_1(X, \mathcal F, \mu, \mathbb R)$. In this case, $$ \int_X f \mathrm d \mu = \int_X g \mathrm d \mu + i \int_X h \mathrm d \mu . $$
- (iii) $f \in L_1(X, \mathcal F, \mu, \mathbb R)$ if and only if $f^+, f^- \in L_1(X, \mathcal F, \mu, \mathbb R_{\ge 0})$ where $f^+, f^-$ are the positive and negative parts of $f$ respectively. In this case, $$ \int_X f \mathrm d \mu = \int_X f^+ \mathrm d \mu - \int_X f^- \mathrm d \mu \quad \text{and} \quad \int_X |f| \mathrm d \mu = \int_X f^+ \mathrm d \mu + \int_X f^- \mathrm d \mu. $$
Now we assume $\mu$ is a probability measure, $f = (f_1, \ldots, f_n) \in L_1(X, \mathcal F, \mu, \mathbb K^n)$, and $\mathcal G$ a sub $\sigma$-algebra of $\mathcal F$. My goal is to apply Corollary 2.12 to define the conditional expectation $\mathbb E (f|\mathcal G)$, i.e.,
- $\mathbb E (f|\mathcal G) \in L_1(X, \mathcal G, \mu, \mathbb K^n)$, and
- $$\int_A f \mathrm d \mu = \int_A \mathbb E (f|\mathcal G) \mathrm d \mu \quad \forall A \in \mathcal G.$$
Could you have a check on my below attempt?
Uniqueness. It follows from this result.
Existence.
a. Let $\mathbb K= \mathbb C$ and $n=1$.
We decompose $f = g +ih$ where $g,h:X \to \mathbb R$. By Corollary 2.12 (ii), $g,h \in L_1(X, \mathcal F, \mu, \mathbb R)$. Decompose $g = g^+ - g^-$. By Corollary 2.12 (iii), $g^+, g^- \in L_1(X, \mathcal F, \mu, \mathbb R_{\ge 0})$. We define a measure $\nu^+ : \mathcal G \to [0, \infty]$ by $$ \nu^+ (A ) := \int_A g^+ \mathrm d \mu \quad \forall A \in \mathcal G. $$
Because $g^+$ is $\mu$-integrable, we get $\nu^+$ is a finite measure. Clearly, $\nu^+ \ll \mu_{\restriction \mathcal G}$. By Radon–Nikodym theorem, there is $\mathbb E (g^+|\mathcal G) \in L_0(X, \mathcal G, \mu, [0, \infty])$ such that $$ \nu^+ (A ) = \int_A \mathbb E (g^+|\mathcal G) \mathrm d \mu \quad \forall A \in \mathcal G. $$
Because $\nu^+$ is finite, $\mathbb E (g^+|\mathcal G) \in L_1 (X, \mathcal G, \mu, \mathbb R_{\ge 0})$. We define $\mathbb E (g^-|\mathcal G)$ similarly. Let $$ \mathbb E (g|\mathcal G) := \mathbb E (g^+|\mathcal G) -\mathbb E (g^-|\mathcal G). $$
Clearly,
- $\mathbb E (g|\mathcal G) \in L_1 (X, \mathcal G, \mu, \mathbb R)$, and
- $$ \int_A g \mathrm d \mu = \int_A \mathbb E (g|\mathcal G) \mathrm d \mu \quad \forall A \in \mathcal G. $$
We define $\mathbb E (h|\mathcal G)$ similarly. Let $$ \mathbb E (f|\mathcal G) := \mathbb E (g|\mathcal G) +i\mathbb E (h|\mathcal G). $$
By Corollary 2.12 (ii),
- $\mathbb E (f|\mathcal G) \in L_1 (X, \mathcal G, \mu, \mathbb C)$, and
- $$ \int_A \mathbb E (f|\mathcal G) \mathrm d \mu = \int_A \mathbb E (g|\mathcal G) \mathrm d \mu + i \int_A \mathbb E (h|\mathcal G) \mathrm d \mu = \int_A g \mathrm d \mu + i\int_A h \mathrm d \mu = \int_A f \mathrm d \mu \quad \forall A \in \mathcal G. $$
b. Let $\mathbb K= \mathbb C$.
By Corollary 2.12 (i), $f_j \in L_1(X, \mathcal F, \mu, \mathbb K)$ for all $j$. The existence of $\mathbb E (f_j|\mathcal G)$ is then guaranteed by part a. above. Let $$ \mathbb E (f|\mathcal G) := \big ( \mathbb E (f_1|\mathcal G), \ldots, \mathbb E (f_n|\mathcal G) \big ). $$
By Corollary 2.12 (i),
- $\mathbb E (f|\mathcal G) \in L_1 (X, \mathcal G, \mu, \mathbb C^n)$, and
- $$ \int_A \mathbb E (f|\mathcal G) \mathrm d \mu = \left ( \int_A \mathbb E (f_1|\mathcal G) \mathrm d \mu, \ldots, \int_A \mathbb E (f_1|\mathcal G) \mathrm d \mu \right ) = \left ( \int_A f_1 \mathrm d \mu, \ldots, \int_A f_n \mathrm d \mu \right ) = \int_A f \mathrm d \mu. $$
c. Let $\mathbb K= \mathbb R$.
We consider $\mathbb R^n$ as a closed subspace of $\mathbb C^n$. Then apply b. to get $\mathbb E (f|\mathcal G) \in L_1 (X, \mathcal G, \mu, \mathbb C^n)$. Notice that $f$ takes values in $\mathbb R^n$ everywhere. Then $\mathbb E (f|\mathcal G)$ takes values in $\mathbb R^n$ almost everywhere by Theorem 1.4. from Rudin's Real and Complex Analysis.
