a square-integrable martingale
A square-integrable martingale is a martingale M with sup_t E[M_t^2] < infinity (equivalently, bounded in L^2). This is the most workable class of martingales: square-integrability is exactly the integrability that makes the predictable quadratic variation <M> exist via Doob-Meyer, makes the Ito isometry an honest equality, and turns the collection of such martingales into a Hilbert space on which the stochastic integral acts as an isometry. It is the comfortable middle ground — more regular than general or local martingales, broad enough to contain Brownian motion and the integrals one cares about.
Precisely, take cadlag martingales M with M_0 = 0 and the norm given by the L^2 norm of the terminal value (or sup_t E[M_t^2]^{1/2}). The space of these, often written M^2 or H^2, is a Hilbert space under the inner product (M, N) = E[M_infinity N_infinity] = E[<M, N>_infinity]; the second equality is the angle-bracket realization of the inner product. Square-integrability gives several equivalent goods at once: M is uniformly integrable (since L^2-boundedness implies UI), so it closes with M_t = E[M_infinity given F_t]; M^2 - <M> is a true martingale, so E[M_t^2] = E[<M>_t] (the L^2 norm equals the expected angle bracket); and the orthogonal decomposition of M^2 into continuous and purely-discontinuous parts holds. Two square-integrable martingales are orthogonal exactly when <M, N> = 0, i.e. MN is a martingale.
Why this class is the workhorse: the Ito integral is first constructed as an isometry from L^2(d<M>) predictable integrands into M^2 — E[(integral H dM)^2] = E[integral H^2 d<M>] — and only then extended by localization to local martingales and semimartingales. Doob's L^2 maximal inequality controls the running supremum, and the BDG inequalities at p = 2 reduce to the bracket identity. The honest qualifications: square-integrability is genuinely stronger than the L^1-boundedness that mere a.s. convergence needs, and stronger than uniform integrability (L^2 implies UI but not conversely); and many central objects — stochastic integrals with merely locally-L^2 integrands, the inverse-Bessel process — are only LOCALLY square-integrable, which is why the locally-square-integrable class and localization are needed to reach the full theory.
Brownian motion B is a square-integrable martingale on any finite horizon: E[B_t^2] = t, and the identity E[B_t^2] = E[<B>_t] = t is the angle-bracket realization of its L^2 norm. The stochastic integral integral_0^t H_s dB_s of a predictable H with E[integral_0^t H^2 ds] < infinity is then another square-integrable martingale with the same isometry.
In M^2 the squared norm equals the expected angle bracket, E[M_t^2] = E[<M>_t] — the basis of the Ito isometry.
Square-integrability implies uniform integrability and closure but is strictly stronger; many integrands and processes are only locally square-integrable, so the full theory still needs localization beyond the M^2 setting.