17.5 Appendix f to applied probability: properties of conditional

We suppose, without repeated assertion, that the random variables and functions of random vectors are integrable, as needed.

• (CE1) Defining condition . $e\left(X\right)=E\left[g\right(Y\left)\right|X]$ a.s. iff $E\left[I_M\left(X\right)g\left(Y\right)\right]=E\left[I_M\left(X\right)e\left(X\right)\right]$ for each Borel set M on the codomain of X .
• (CE1a) If $P(X\in M)>0$ , then $E\left[I_M\left(X\right)e\left(X\right)\right]=E\left[g\left(Y\right)\right|X\in M]P(X\in M)$
• (CE1b) Law of total probability . $E\left[g\right(Y\left)\right]=E\left\{E\right[g\left(Y\right)\left|X\right]\}$
• (CE2) Linearity . For any constants $a,b$
  $$E\left[ag\right(Y)+bh(Z\left)\right|X]=aE[g\left(Y\right)\left|X\right]+bE\left[h\right(Z\left)\right|X]a.s.$$
  (Extends to any finite linear combination)
• (CE3) Positivity; monotonicity .
  1. $g\left(Y\right)\ge 0$ a.s. implies $E\left[g\right(Y\left)\right|X]\ge 0$ a.s.
  2. $g\left(Y\right)\ge h\left(Z\right)$ a.s. implies $E\left[g\right(Y\left)\right|X]\ge E[h\left(Z\right)\left|X\right]$ a.s.
• (CE4) Monotone convergence . $Y_n\to Y$ a.s. monotonically implies $$E\left[Y_n\right|X]\to E\left[Y\right|X]a.s.$$
• (CE5) Independence . $\{X,Y\}$ is an independent pair
  • iff $E\left[g\right(Y\left)\right|X]=E[g\left(Y\right)]$ a.s. for all Borel functions g
  • iff $E\left[I_N\left(Y\right)\right|X]=E\left[I_N\left(Y\right)\right]$ a.s. for all Borel sets N on the codomain of Y
• (CE6) $e\left(X\right)=E\left[g\right(Y\left)\right|X]$ a.s. iff $$E\left[h\right(X\left)g\right(Y\left)\right]=E\left[h\right(X\left)e\right(X\left)\right]a.s.\text{for}\text{any}\text{Borel}\text{function}h$$
• (CE7) $E\left[h\right(X\left)\right|X]=h(X)$ a.s. for any Borel function h
• (CE8) $E\left[h\right(X\left)g\right(Y\left)\right|X]=h(X\left)E\right[g\left(Y\right)\left|X\right]$ a.s. for any Borel function h
• (CE9) If $X=h\left(W\right)$ , then $E\left\{E\right[g\left(Y\right)\left|X\right]\left|W\right\}=E\left\{E\right[g\left(Y\right)\left|W\right]\left|X\right\}=E\left[g\right(Y\left)\right|X]$ , a.s.
• (CE9a) $E\left\{E\right[g\left(Y\right)\left|X\right]|X,Z\}=E\left\{E\right[g\left(Y\right)|X,Z]\left|X\right\}=E\left[g\right(Y\left)\right|X]$ a.s.
• (CE9b) If $X=h\left(W\right)$ and $W=k\left(X\right)$ , with $h,k$ Borel functions, then $$E\left[g\right(Y\left)\right|X]=E[g\left(Y\right)\left|W\right]a.s.$$
• (CE10) If g is a Borel function such that $E\left[g\right(t,Y\left)\right]$ is finite for all t on the range of X and $E\left[g\right(X,Y\left)\right]$ is finite, then
  1. $E\left[g\right(X,Y\left)\right|X=t]=E[g(t,Y)|X=t]$ a.s. $\left[P_X\right]$
  2. If $\{X,Y\}$ is independent, then $E\left[g\right(X,Y\left)\right|X=t]=E[g(t,Y)]$ a.s. $\left[P_X\right]$
• (CE11) Suppose $\left\{X\right(t):t\in T\}$ is a real-valued, measurable random process whose parameter set T is a Borel subset of the real line and S is a random variable whose range is a subset of T , so that $X\left(S\right)$ is a random variable.
  If $E\left[X\right(t\left)\right]$ is finite for all t in T and $E\left[X\right(S\left)\right]$ is finite, then
  1. $E\left[X\right(S\left)\right|S=t]=E[X\left(t\right)|S=t]$ a.s. $\left[P_S\right]$
  2. If, in addition, $\{S,X_T\}$ is independent, then $E\left[X\right(S\left)\right|S=t]=E[X\left(t\right)]$ a.s. $\left[P_S\right]$
• (CE12) Countable additivity and countable sums .
  1. If Y is integrable on A and ${\displaystyle A=\underset{n=1}{\overset{\infty }{\bigvee }}{A}_n}$ ,
     then ${\displaystyle E\left[I_{A}Y\right|X]=\sum _{n=1}^{\infty }E\left[I_{A_n}Y\right|X]}$ a.s.
  2. If ${\displaystyle \sum _{n=1}^{\infty }E\left[\right|Y_n\left|\right]<\infty }$ , then ${\displaystyle E\left[\sum _{n=1}^{\infty },Y_n,|X\right]=\sum _{n=1}^{\infty }E\left[Y_n\right|X]}$ a.s.
• (CE13) Triangle inequality . $\left|E\right[g\left(Y\right)\left|X\right]|\le E[\left|g\right(Y\left)\right|\left|X\right]$ a.s.
• (CE14) Jensen's inequality . If g is a convex function on an interval I which contains the range of a real random variable Y , then $$g\left\{E\right[Y\left|X\right]\}\le E[g\left(Y\right)\left|X\right]a.s.$$
• (CE15) Suppose $E\left[\right|Y{|}^p]<\infty$ and $E\left[\right|Z{|}^p]<\infty$ for $1\le p<\infty$ . Then $${E\left\{\right|E\left[Y\right|X]-E\left[Z\right|X]|}^p{\}\le E[|Y-Z|}^p]<\infty$$