I would like to check my understanding of some facts related to conditional expectations.

We know that a conditional expectation of a non-negative or integrable r.v. $X$ (defined on $(\Omega,\mathcal A, P)$) w.r.t. $\mathcal C \subset \mathcal A$ is another random variable $X_0$ satisfying certain well-known conditions ($\star$). Moreover there is no guarantee that only $X_0$ does satisfy ($\star$). But what we can guarantee is that if $\hat{X}_0$ also satisfies ($\star$), then $X_0 = \hat{X}_0$ a.s.

  1. $G(X,\mathcal C)$ is a set of conditional expectations for $X$, can we find $A \in \mathcal A$ such that on $A$ all elements of $G(X,\mathcal C)$ are equal to each other? (I do not know)

  2. If the answer to the previous question is no, then would it change if instead of $G(X,\mathcal C)$ we work with a countable subset of $G(X,\mathcal C)$? (I think yes, because a countable union of null-sets is a null set)

  3. If $X_0 \in G(X,\mathcal C)$ and $\hat{X}_0 := X_0 * I_{N^c} + a*I_{N}$, $N \in \mathcal C$, $a \in \mathbb R_0^+ \bigcup +\infty$ if $X \ge 0$ or $a \in \mathbb R \bigcup \pm \infty$ if $X$ integrable, $P(N)=0$, would it follow that $\hat{X}_0 \in G(X,\mathcal C)$? (I think yes, because $\hat{X}_0$ is still $\mathcal C$ measurable, non-negative or integrable and has the same value of $P$-integral over any $C \in \mathcal C$)

  4. What if in the question above $a$ is a $\mathcal C$ measurable numeric function on $\Omega$ (which is non-negative if $X \ge 0$)? (I think the answer is still yes)

  • 1
    $\begingroup$ No, if G is uncountable. Yes. Yes. Yes, without the condition that the random variable $A$ is nonnegative. $\endgroup$ – Did Sep 20 '15 at 11:43

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