Inf Closed mem countable Inf Closure iff
InfClosed.mem_countableInfClosure_iff
Plain-language statement
If the set is inf-closed, elements of countablInfClosure can be written as countable intersections of antitone sequences of sets.
Exact Lean statement
lemma _root_.InfClosed.mem_countableInfClosure_iff (hp : InfClosed p) {s : Set 𝓧} :
s ∈ countableInfClosure p ↔ ∃ A : ℕ → Set 𝓧, (∀ n, A n ∈ p) ∧ Antitone A ∧ s = ⋂ n, A nFormal artifact
Lean source
lemma _root_.InfClosed.mem_countableInfClosure_iff (hp : InfClosed p) {s : Set 𝓧} : s ∈ countableInfClosure p ↔ ∃ A : ℕ → Set 𝓧, (∀ n, A n ∈ p) ∧ Antitone A ∧ s = ⋂ n, A n := by rw [mem_countableInfClosure_iff_iInf] refine ⟨fun h ↦ ?_, fun ⟨A, hA, _, h_eq⟩ ↦ ⟨A, hA, h_eq.symm⟩⟩ choose A hA hs using h refine ⟨Set.dissipate A, fun n ↦ ?_, Set.antitone_dissipate, ?_⟩ · induction n with | zero => simp [hA] | succ n hn => rw [Set.dissipate_succ] exact hp hn (hA _) · rw [Set.iInter_dissipate, ← hs] rfl- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/Choquet/CompactSystem.lean:352-364
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.