uniform integrability of all conditional expectations of a fixed $L^1$ function

Question

Let $Z$ be a real $L^1$ random variable on a probability space $(\Omega, \mathscr{A}.\mu).$ Why is the family of all $E[Z| \mathscr{B}]$ uniformly integrable when $ \mathscr{B}$ ranges over the sub-sigma-algebras of $ \mathscr{A}$? This is supposed to be trivial but even assuming $Z\geq 0$ i don't know how to control $\int_A E[Z| \mathscr{B}]d\mu$ when $A\notin \mathscr{B}$.

Answer 1

Recall that because $Z$ is $L^1$, for any $\epsilon > 0$, there is a $\delta > 0$ such that $E[|Z|;A] < \epsilon$ whenever $\mu(A) < \delta$. Also we use an equivalent definition of uniform integrability that a family $\{X_i\}$ of r.v.s is u.i. if $\lim_{M\to\infty}\sup_iE[|X_i|;|X_i|>M] = 0$.

Now, for any sub-$\sigma$-algebra $\mathscr B$ of $\mathscr A$, let $X$ be a version of $E[Z|\mathscr B]$ and $Y$ be a version of $E[|Z|\mid\mathscr B]$. Then, \begin{align*} E[|X| ; |X|>M] &\le E[Y;Y>M]\\ &= E[|Z|; Y > M]. \end{align*} Now, $\mu(Y > M) \le \frac{1}{M}E[Y] = \frac{1}{M}E[|Z|]$. If we choose $M$ so large that $E[|Z|] < M\delta$, then $\mu(Y>M) < \delta$, so $E[|X| ; |X|>M] < \epsilon$.