Limsup le of eventually monotone of tendsto on dense
limsup_le_of_eventually_monotone_of_tendsto_on_dense
Plain-language statement
Convergence on a dense set of a collection of monotone function controls the limsup at a point if f is right continuous at a. We prove this under the assumption that α has both a bottom element and a top element. The bottom element is needed because otherwise limsup evaluated at the bottome element may give a junk value to break the inequality.
Exact Lean statement
lemma limsup_le_of_eventually_monotone_of_tendsto_on_dense {ι α β : Type*} [LinearOrder α]
[BoundedOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {l : Filter ι}
[l.NeBot] {D : Set α} {F : ι → α → β} {f : α → β} (hF : ∀ᶠ i in l, Monotone (F i))
(hD : Dense D) (htop : ⊤ ∈ D) (hbot : ⊥ ∈ D) {a : α} (hfa : ContinuousWithinAt f (Set.Ioi a) a)
(hlim : ∀ t ∈ D, Tendsto (F · t) l (𝓝 (f t))) :
limsup (F · a) l ≤ f aFormal artifact
Lean source
lemma limsup_le_of_eventually_monotone_of_tendsto_on_dense {ι α β : Type*} [LinearOrder α] [BoundedOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {l : Filter ι} [l.NeBot] {D : Set α} {F : ι → α → β} {f : α → β} (hF : ∀ᶠ i in l, Monotone (F i)) (hD : Dense D) (htop : ⊤ ∈ D) (hbot : ⊥ ∈ D) {a : α} (hfa : ContinuousWithinAt f (Set.Ioi a) a) (hlim : ∀ t ∈ D, Tendsto (F · t) l (𝓝 (f t))) : limsup (F · a) l ≤ f a := by by_cases! ha : a = ⊤ · rw [ha, (hlim ⊤ htop).limsup_eq] · have : (comap ((↑) : D → α) (𝓝[>] a)).NeBot := hD.comap_val_nhdsWithin_Ioi_neBot ha.lt_top refine (isClosed_Ici (a := limsup (F · a) l)).mem_of_tendsto (Tendsto.comp hfa (tendsto_comap (f := ((↑) : D → α)))) ?_ rw [eventually_comap, eventually_nhdsWithin_iff] filter_upwards with z hz d rfl simp only [Function.comp_apply, Set.mem_Ici, ← (hlim d d.2).limsup_eq] refine limsup_le_limsup ?_ ?_ (hlim d d.2).isBoundedUnder_le · filter_upwards [hF] with i hi using hi hz.le · refine (hlim ⊥ hbot).isCoboundedUnder_le.trans ?_ filter_upwards [hF] with i hi using hi bot_le- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:971-989
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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.
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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.