Change the order of conditional expectation of integration

Question

I encountered this problem when learning SDE:

$g(t,\omega)$ is a adapted process

then $$\mathbb E\left(\int_a^b |g(t)|^2 \, dt \mid \mathcal F_a\right)=\int_a^b\mathbb E\left(|g(t)|^2\mid \mathcal F_a\right) \, dt$$

I don't know whether I can change the order of conditional expectation of integration.

So I try to prove it using the definition of conditional expectation but failed, does it need additional condition?

Answer 1

The claim follows from Tonelli's theorem. Recall that a random variable $Y \in L^1(\mathcal{F})$ equals $\mathbb{E}(X \mid \mathcal{F})$ if, and only if,

$$\int_F Y \, d\mathbb{P} = \int_F X \, d\mathbb{P} \tag{1}$$

for all $F \in \mathcal{F}$. Set $X:= \int_a^b |g(t)|^2 \, dt$. Since, by $(1)$,

$$\int_F \mathbb{E}(|g(t)|^2 \mid \mathcal{F}_a) \, d\mathbb{P} = \int_F |g(t)|^2 \, d\mathbb{P}$$

we get

$$\int_a^b \left( \int_F \mathbb{E}(|g(t)|^2 \mid \mathcal{F}_a) \, d\mathbb{P} \right) dt = \int_a^b \left( \int_F |g(t)|^2 \, d\mathbb{P} \right) \, dt. \tag{2}$$

By Tonelli's theorem,

$$\begin{align*} \int_F X \, d\mathbb{P} &= \int_a^b \int_F |g(t)|^2 \, d\mathbb{P} \, dt \stackrel{(2)}{=} \int_F\left( \int_a^b \mathbb{E}(|g(t)|^2 \mid \mathcal{F}_a) \, dt \right) \, d\mathbb{P} \end{align*}$$

for all $F \in \mathcal{F}_a$, i.e. $Y := \int_a^b \mathbb{E}(|g(t)|^2 \mid \mathcal{F}_a) \, dt$ satisfies $(1)$.