Construction

Let $(Y_i)_{i\ge 1}$ be i.i.d. random variables such that

$$ \mathbb E[|Y_1|]<\infty, \qquad \mathbb E[Y_1]=0. $$

Define

$$ X_n=\sum_{i=1}^n Y_i, \qquad \mathcal F_n=\sigma(Y_1,\dots,Y_n). $$

Then $(X_n)$ is a martingale with respect to $(\mathcal F_n)$.

Verification

Since $X_n$ is determined by $Y_1,\dots,Y_n$, it is $\mathcal F_n$-measurable.

Also,

$$ \mathbb E[|X_n|]\le \sum_{i=1}^n \mathbb E[|Y_i|]<\infty. $$

For the martingale property,

X_n+\mathbb E[Y_{n+1}\mid \mathcal F_n]. $$

Because $Y_{n+1}$ is independent of $\mathcal F_n$,

$$ \mathbb E[Y_{n+1}\mid \mathcal F_n]=\mathbb E[Y_{n+1}]=0. $$

Hence

$$ \mathbb E[X_{n+1}\mid \mathcal F_n]=X_n. $$