English
If S is a family of sets and every chain has an upper bound not equal to top, then there is a minimal element in S.
Русский
Если S — семейство множеств, и каждая цепь имеет верхнюю грань не равную ⊤, то существует минимальный элемент в S.
LaTeX
$$$$\forall S, \Big(\forall c \subseteq S, IsChain (\\subseteq) c \to \exists ub \in S, \forall s \in c, ub \subseteq s\Big) \to \exists m, \ Minimal (\\in S) m.$$$$
Lean4
theorem indep_comap_natural_of_lt (hf : ∀ i, StronglyMeasurable (f i)) (hfi : iIndepFun f μ) (hij : i < j) :
Indep (MeasurableSpace.comap (f j) mβ) (Filtration.natural f hf i) μ :=
by
suffices
Indep (⨆ k ∈ ({ j } : Set ι), MeasurableSpace.comap (f k) mβ) (⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ
by rwa [iSup_singleton] at this
exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)
