I'm a little bewildered on how to get the following proved...
Suppose we make the following assumptions:
Let $f_n$ be a measurable function on $\mathbb{R}^n$. Let $Z_1,Z_2,\cdots$ be independant random variables and $\mathcal{F}_n = \sigma(Z_1,\cdots, Z_n)$. Let $X_n = f_n(Z_1,\cdots, Z_n)$ and assume that $\mathbb{E}|X_n|<\infty$ and $\mathbb{E}f_n(z_1,\cdots, z_{n-1}, Z_n) = f_{n-1}(z_1,\cdots, z_{n-1})$ for all $n$.
Apparantly then $X_n$ defines a martingale...I just feel like i'm missing some basic understanding about martingales to see why $\mathbb{E}[X_{n+1}|\mathcal{F}_n] = X_n$
holds in this case. Would someone be so nice to enlighten m