the usual conditions on a filtration
A filtration is a growing family of sigma-algebras (F_t) on a probability space, indexed by continuous time t >= 0, where F_t records everything observable up to time t. In discrete time you can be cavalier about exactly which events sit in F_t, but in continuous time the index set is uncountable and naive definitions break: events like 'the path has hit level a by time t' or 'the supremum over s <= t exceeds c' need not be measurable, and limits along the uncountably many times before t can leak out of F_t. The usual conditions are two technical regularity demands that tame these pathologies and are assumed in virtually all of continuous-time martingale theory.
Concretely, a filtration satisfies the usual conditions if it is complete (every F_t contains all the P-null sets of the whole space, so that almost-sure statements transfer cleanly and modifications of processes stay adapted) and right-continuous (F_t equals the intersection over s > t of F_s, written F_t = F_{t+}, so that 'what you know an instant after t' is already known at t). Given any raw filtration generated by a process, one forms its usual augmentation by first adding all null sets and then taking right limits; the result is the smallest filtration containing the original that satisfies the usual conditions.
Why insist on this? Right-continuity is exactly what lets you choose cadlag (right-continuous with left limits) modifications of martingales, makes hitting times of open and closed sets into stopping times, and underlies the Debut theorem and the optional section theorems. Completeness keeps adaptedness stable under almost-sure equality. The price is mild: completion enlarges the sigma-algebras only by null sets, and for a Feller process or Brownian motion the augmented filtration is automatically right-continuous (Blumenthal's 0-1 law is the F_0 = F_{0+} statement). The honest caveat is that the usual conditions are a convenience, not a triviality: dropping right-continuity genuinely destroys the optional stopping and Doob-Meyer machinery, and some modern treatments work hard to avoid completion when they need the raw, possibly non-complete, filtration.
Take Brownian motion B started at 0 and let F_t be the sigma-algebra it generates. The raw F_0 contains only trivial events, but Blumenthal's 0-1 law says the augmented F_{0+} is also trivial (every event in it has probability 0 or 1) — which is exactly why F_t is right-continuous after augmentation, and why a statement like 'B immediately enters the positive half-line' has a definite probability.
Blumenthal's 0-1 law is the statement that the usual augmentation of Brownian filtration already satisfies F_0 = F_{0+}.
Completeness and right-continuity are independent demands; a filtration can have one without the other, and you generally must impose both before quoting the continuous-time optional stopping or Doob-Meyer theorems.