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Conditional expectation

Theorem (Kolmogorov 1933) Let $X$ be a real-valued random variable on $(\Omega,{\mathcal F},P)$ such that E$|X|<\infty$. Let ${\mathcal G}\subset{\mathcal F}$ (${\mathcal G}$ is a sub-$\sigma$-algbra of ${\mathcal F}$), then there exists a random varible $Y$ such that
i) $Y$ is a random variable on $(\Omega,{\mathcal G},P)$ ($Y$ is ${\mathcal G}$-measurable)
ii) E$|Y|<\infty$
iii) For every set $G\in{\mathcal G}$ we have \[ \int_G YdP=E[YI_G]=E[XI_G]=\int_G XdP.\] If $\tilde{Y}$ is another random variable satisfying i)--iii) then $P(Y=\tilde{Y})=1$.

Properties of conditional expectation

Assume that $X$ is a random varible on $(\Omega,{\mathcal F},P)$ with E$|X|<\infty$.
a) If $X$ is ${\mathcal G}$-measurable then $E[X|{\mathcal G}]=X$.
b) If $X$ is independent of ${\mathcal G}$ then $E[X|{\mathcal G}]=E[X]$.
c) (Law of total expectation) $E[X]=E[E[X|{\mathcal G}]]$
d) (Tower property) If ${\mathcal F}_1\subset{\mathcal F}_2\subset{\mathcal F}$ then \[ E[E[X|{\mathcal F}_2]|{\mathcal F}_1]=E[X|{\mathcal F}_1]\] \[E[E[X|{\mathcal F}_1]|{\mathcal F}_2]=E[X|{\mathcal F}_1] \] e) (Taking out what is known) If $Z$ is ${\mathcal G}$-measurable and $E|ZX|<\infty$ then \[ E[ZX|{\mathcal G}]=ZE[X|{\mathcal G}] \] f) (Jensen) If $f$ is a convex (concave) function such that $E|f(X)|<\infty$ then \[ E[f(X)|{\mathcal G}]\stackrel{\geq}{\scriptsize(\leq)}f(E[X|{\mathcal G}])\]