Prove/Disprove $E[X|\mathscr G] = X$ if $X$ is almost surely constant.

Question

$E[X|\mathscr G]$ is any (bounded or integrable or something) random variable $Z$ s.t.

1. $Z$ is $\mathscr G-$measurable
2. $$E[X1_G] = E[Z1_G]$$

If $Z=X$ and $X=c1_A$ where $P(A)=1$, then it looks like $Z=X$ has $(2)$ covered but not $(1)$. If that's the case, please give a counterexample. Otherwise, how does it have $(1)$ covered?

Answer 1

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ is a $\sigma$-field. Suppose $X=c$ almost surely. Then $E(X|\mathcal{G})=c$ almost surely.

Proof. All we need to show is that the constant variable $Y=c$ is a version of $E(X|\mathcal{G})$.

• First of all, $Y$ is $\mathcal{G}$ measurable.
• For any $A\in\mathcal{G}$, we have $$ \int_Ac\ dP=\int_A X\ dP $$ since $X=c$ a.s.

Q.E.D

---

[Added:] Note that the equality $E(X|\mathcal{G})=X$ should be understood as $$ E(X|\mathcal{G})=X\quad a.s. $$ which means $X$ is equal to a version of $E(X|\mathcal{G})$ almost surely. But $X$ itself does not have to be a version of $E(X|\mathcal{G})$ and thus does not have to be $\mathcal{G}$-measurable.

---

[Added later] Suppose $(\Omega,\mathcal{F},P)$ is a probability space and $X=c$ almost surely. Then by definition $P(X\neq c)=0$ and if one defines $$ A=\{\omega\in\Omega\mid X(\omega)= c \} $$ then indeed, $P(A)=1$ and $X=c1_A$ almost surely. But there is no guarantee that $X=c1_A$, (which means $X(\omega)=c1_A(\omega)$ for every $\omega\in\Omega$, )because although $X=c$ on $A$, $X$ could be not zero on the complement of $A$.

Answer 2

Here is a sequence of subquestions, can you answer these? I suspect that your work on these subquestions will make any of your residual confusions disappear. I think the necessary hints for these subquestions have already been given in one or more comments above. So, this will also be a good chance for you to process those comments.

---

Let $\Omega=[0,1]$. Let $\mathcal{F}$ be the usual Borel sigma-algebra and let $P$ be the usual probability measure that defines, for each set $B \in \mathcal{F}$, the probability $P[B]$ equal to the usual measure of set $B$ on the unit interval. Define the trivial sub-sigma algebra $\mathcal{G} = \{\phi, \Omega\}$.

a) Give an example of a set $A \in \mathcal{F}$ such that $P[A]=1$ but $A \neq \Omega$.

b) Fix $c>0$. Let $A$ be the set from part (a). Define the random variable $X = c 1_A$. Argue that $X$ is not $\mathcal{G}$-measurable.

c) Conclude that the random variable $X$ of part (b) is not a version of $E[X|\mathcal{G}]$. [I believe this concludes your original question.]

d) Argue that $X=c$ almost surely.

e) Define the constant random variable $Z=c$. Prove that $Z$ is a version of $E[X|\mathcal{G}]$.

f) Argue that if $W$ is a version of $E[X|\mathcal{G}]$, then $W=c$ almost surely.

g) [related to the last d.k.o. comment] Argue that if $W$ is a version of $E[X|\mathcal{G}]$, then the event $\{W \neq c\}$ is in $\mathcal{G}$. Also, $P[W \neq c]=0$.