Basic Properties of Conditional Expectation for Square-Integrable Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $X,X'$ be square-integrable random variables on it. Let $Y$ be a conditional expectation of $X$ given $\mathcal{G}$ and $Y'$ a conditional expectation of $X'$ given $\mathcal{G}$. All parts below are asserted for every such choice of $Y$ and $Y'$. Then:

1. (Linearity) For all real $a,b$, the random variable $aY+bY'$ is a conditional expectation of $aX+bX'$ given $\mathcal{G}$.

2. (Monotonicity) If $P(X\le X')=1$, then $P(Y\le Y')=1$.

3. (Tower property) Let $\mathcal{H}$ be a sub-$\sigma$-algebra of $\mathcal{F}$ with $\mathcal{H}\subseteq\mathcal{G}$. Then every conditional expectation of $Y$ given $\mathcal{H}$ is a conditional expectation of $X$ given $\mathcal{H}$.

4. (Taking out what is known) Let $Z$ be a $\mathcal{G}$-measurable random variable that is bounded, meaning there is a real $C$ with $|Z(\omega)|\le C$ for every $\omega$. Then $ZX$ and $ZY$ are square-integrable, and $ZY$ is a conditional expectation of $ZX$ given $\mathcal{G}$.

5. (Independence) If the $\sigma$-algebras $\sigma(X)$ and $\mathcal{G}$ are independent, then the constant random variable with value $\mathbb{E}[X]$ is a conditional expectation of $X$ given $\mathcal{G}$; consequently $P\bigl(Y=\mathbb{E}[X]\bigr)=1$.

6. (Mean-square contraction) $\lVert Y\rVert_{2}\le\lVert X\rVert_{2}$, with the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product.