Plain-English statement
At time t, a stopping rule may only use information already present in the filtration at t.
Mathematical statement
For every t >= 0, the event {tau <= t} is F_t-measurable.
Intuition
The rule cannot look into the future to decide whether it has already stopped.
Conditions
- a measurable sample space
- a filtration indexed by nonnegative real time
- an extended nonnegative random time
Why these conditions cannot be dropped
- the filtration records the information available at each time
- WithTop permits a path that never reaches the stopping condition
- event measurability is stronger and more operational than ambient measurability of tau alone
Proof route
- reuse Mathlib's IsStoppingTime predicate exactly
- verify constant stopping times through Mathlib.isStoppingTime_const
Lean interface notes
- NNReal is used for the nonnegative continuous-time index
- the wrapper retains compatibility with Mathlib's stoppedProcess theorems
- no localization convergence theorem is bundled into this definition
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Lean statement
def IsChewiStoppingTime
{Ω : Type*} {m : MeasurableSpace Ω}
(filtration : Filtration ℝ≥0 m) (τ : Ω → WithTop ℝ≥0) : Prop :=
MeasureTheory.IsStoppingTime filtration τ
/-- Constant nonnegative times satisfy the source stopping-time definition. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:27published source at 7bcd37294df1