# Reconciling Definitions of Conditional Expectation

In Risk and Portfolio Analysis (Hult et al, 2014) chapter 3.1 conditional expectation is defined as follows:

Consider a random vector $$Z$$ and a random variable $$L$$ with $$\mathbb{E}[L^2] < \infty$$. The conditional expectation of $$L$$ given $$Z$$, written $$\mathbb{E}[L \mid Z]$$, is the random variable $$g(Z)$$ with $$g: \mathbb{R}^n \to \mathbb{R}$$ such that $$\mathbb{E}[g(Z)^2] < \infty$$ satifying $$\mathbb{E}[h(Z)g(Z)] = \mathbb{E}[h(Z)L]$$ for all $$h: \mathbb{R}^n \to \mathbb{R}$$ such that $$\mathbb{E}[h(Z)^2] < \infty$$.

I am having a hard time reconciling this definition with the abstract version (as stated for instance on wikipedia):

Let $$(\Omega, \mathcal{F}, \mathbb{P})$$ be a probability space and $$\mathcal{G} \subseteq \mathcal{F}$$ be a sub-$$\sigma$$-algebra. If $$X: \Omega \to \mathbb{R}$$ is a random variable then the conditional expectation of $$X$$ given $$Y$$ is a $$\mathcal{G}$$-measurable function $$Y$$ such that for all $$G\in \mathcal{G}$$ \begin{align*} \int_G Y d\mathbb{P} = \int_G X d\mathbb{P}. \end{align*} We write $$Y$$ as $$\mathbb{E}[X \mid \mathcal{ G}]$$. If we let $$\chi$$ denote the characteristic function we can also write $$\mathbb{E}[\chi_G\mathbb{E}[ X \mid \mathcal{G}]] = \mathbb{E}[\chi_G X].$$

I would like to know how you can establish the equivalence (if it is an equivlance at all) between the two definitions? I'm thinking that there should be a way to go from the second to the first.

In the second definition take $$\mathcal G=\sigma(Z)=\{Z^{-1}(E): E \,\, \text {Borel in } \mathbb R\}$$. If the equation in the first definition holds, you can take $$h=1_E$$ to get the equation in the second . In the converse direction, you have to start with $$h=1_E$$, then go to simple functions, and then to arbitrary meaurable functions $$h$$ with $$E[h(Z)]^{2}<\infty$$.
• Ahh okay! I thought about taking $h = 1_E$ to go from the second to the first but not far enough that it implies that it should hold for any $h$ square-integrable. Is it shown using the "usual" way, i.e. showing it for positivie measurable functions first using some kind of montone convergence theorem? Sep 29 at 12:24
• Yes, on $(\mathbb R,P\circ Z^{-1})$ you can approximate any $L^{2}$ functions by sequences of simple functions in the standard way. @Pakchoiandbacon Sep 29 at 12:28