My lecture note defines conditional expectation as follows:

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{G}$ be a sub-sigma field of $\mathcal{F}$. If $X$ is an integrable random variable, then the conditional expectation of $X$ given $\mathcal{G}$ is any integrable random variable $Z$ which satisfies the following two properties:

(CE1) $Z$ is $\mathcal{G}$-measurable.

(CE2) $$\forall \Lambda \in \mathcal{G}: \int_{\Lambda} Z \, d \mathbb{P}=\int_{\Lambda} X \, d \mathbb{P}$$

We denote $Z$ by $\mathbb{E}[X | \mathcal{G}]$.

Let $X$ be an integrable random variable and $\mathcal G$ a sub-sigma field.

I would like to ask if we can deduce a formula (which maybe related to integral or probability) of $\mathbb{E}[X | \mathcal{G}] (\omega)$ from (CE1) and (CE2).

Thank you so much!

  • 1
    $\begingroup$ In general, $\mathbb{E}[X | \mathcal{G}]$ is defined only up to sets of measure zero. If singleton $\{\omega\}$ has measure zero, then changing $\mathbb{E}[X | \mathcal{G}]$ on $\omega$ makes no difference. If you have studied measure theory, you may like to think of $\mathbb{E}[X | \mathcal{G}]$ as an example of a Radon-Nikodym derivative. $\endgroup$ – GEdgar Jan 27 '20 at 22:15
  • $\begingroup$ Hi @GEdgar, if $X$ is the indicator function $\mathbb{1}_\Lambda$, can we have a formula for $\mathbb{E}[\mathbb{1}_\Lambda | \mathcal{G}] (\omega)$ without Radon-Nikodym derivative? $\endgroup$ – LE Anh Dung Jan 27 '20 at 22:45

If $\mu(\cdot,\cdot)$ is the regular conditional distribution of $X$ given $\mathcal{G}$, then for an integrable function $g$, $$ \mathsf{E}[g(X)\mid \mathcal{G}](\omega)=\int_{\mathbb{R}} g(x)\mu(\omega,dx) \quad\text{a.s.} $$

  • $\begingroup$ Hi @d.k.o, if $X$ is the indicator function $\mathbb{1}_\Lambda$, can we have a formula for $\mathbb{E}[\mathbb{1}_\Lambda | \mathcal{G}] (\omega)$ without regular conditional distribution? $\endgroup$ – LE Anh Dung Jan 27 '20 at 22:46
  • $\begingroup$ What kind of formula are you looking for? $\endgroup$ – d.k.o. Jan 27 '20 at 22:46

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