Square-integrable martingale ($\mathbb{E}|M_t{|}^2<\infty \text{ for every finite }t$)

A martingale is square integrable when its value at every time belongs to L2 space. This implies ordinary integrability by Cauchy-Schwarz inequality. The condition is separate from a uniform bound on these second moments over an infinite horizon. An Itô integral is square integrable on a finite interval when its predictable integrand has finite expected integrated square, by Itô isometry.