Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk

Question

Prove that $M(t)^2 - t$ is a martingale, $M(t)$ is a symmetric random walk.

My question here mainly has to do with the $F_{t}$ measurability of $M(t)^2 - t$, where $F_{t} = \sigma (X_1 , X_2, ... , X_t)$.

If a function $M(t)$ is a measurable wrt $F_{t} = \sigma (X_1 , X_2, ... , X_t)$, is it always true that its square is also $F_{t}$- measurable?

Answer 1

Indeed, the composition of measurable functions remains measurable. In your case, both, $M(t)$ and $x\to x²$ are measurable.