Suppose $\{X_t\}_{t=1}^{\infty}$ is a sequence of random variables. Let $I_t$ denote the information set at time $t$. $E_t[X_{t+1}]$ is short for $E[X_{t+1} \mid I_t]$.
Next, assume $E_t[X_{t+1}] = 0$. Is the following claim true?
Suppose $Z_t \subset I_t$. $E[X_{t+1} \mid Z_t] = 0$.
I don't think it is, but I'm having a hard time finding a counter-example. Can I get a hint?
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and let $\mathcal G_1$ and $\mathcal G_2$ be two sub-$\sigma$-algebras of $\mathcal F$ such that $\mathcal G_1\subset \mathcal G_2$. If $X$ is an integrable random variable such that $\mathbb E\left[X\mid\mathcal G_2\right]=0$ almost surely, then $\mathbb E\left[X\mid\mathcal G_1\right]=0$ almost surely.
Indeed, for each $G_1\in\mathcal G_1$ we have $G_1\in\mathcal G_2$ hence $\mathbb E\left[X\mathbf{1}_{G_1}\right]=\mathbb E\left[\mathbb E\left[X\mid\mathcal G_2\right]\mathbf{1}_{G_1}\right]=0$.