the predictable sigma-algebra
Continuous-time stochastic integration cannot integrate against just any adapted process: it must use integrands that are knowable 'just before' each instant, not at the instant. The predictable sigma-algebra is the rigorous formalization of 'knowable an instant in advance', and it is the class of integrands for which the Ito integral, the Doob-Meyer compensator, and the predictable quadratic variation are well-defined. Alongside it sits a slightly larger sigma-algebra, the optional sigma-algebra, of processes knowable 'at the instant'; the gap between them is invisible for continuous processes but decisive for jumps.
Formally, a process is viewed as a function of (omega, t) on the product space Omega x [0, infinity). The predictable sigma-algebra P is the sigma-algebra on this product generated by all left-continuous adapted processes — equivalently, generated by the 'predictable rectangles' A x (s, t] with A in F_s. A process is predictable if it is P-measurable; the prototypes are left-continuous adapted processes and, more generally, limits of such. The optional sigma-algebra O is generated instead by the cadlag (right-continuous) adapted processes and contains P; an optional process is adapted and cadlag-built. The intuition: a left-continuous integrand evaluated at time t uses information strictly before t (value is the left limit), so it is 'predictable'; a cadlag process can jump exactly at t and so encodes information at t. For a predictable process H, the value H_T at a stopping time T is F_{T-}-measurable.
Why this matters: the stochastic integral against a martingale is built only for predictable integrands, and this is not pedantry — allowing the integrand to react to the very jump it multiplies would let you 'see the future' and break the martingale property (you could integrate dM against the sign of dM and manufacture a positive drift). The Doob-Meyer decomposition produces a PREDICTABLE increasing process precisely so it is unique; the predictable quadratic variation <M> is the predictable compensator. The honest subtlety is exactly the predictable-versus-optional distinction at jump times: a jump of a martingale is, by the martingale property, unpredictable in a precise sense (its predictable projection is 0), which is why the compensator must be predictable; conflating the two sigma-algebras is the classic beginner error in jump-process calculus.
A simple predictable integrand is H_t = sum H_i 1_{(t_i, t_{i+1}]}(t) with H_i in F_{t_i}: a step function whose height over (t_i, t_{i+1}] was already chosen at the left endpoint t_i. Integrating such an H against a martingale M and getting back a martingale is the base case from which the whole Ito integral is built; using F_{t_{i+1}}-measurable heights would break it.
Predictability is the 'no peeking at the next move' condition; it is what keeps the stochastic integral a martingale.
Predictable is strictly smaller than optional, and the difference lives entirely at jumps; for continuous processes the distinction is immaterial, but for jump processes choosing the optional version of an integrand can destroy the martingale property.