Integrals against peak functions

A sequence of peak functions is a sequence of functions with average one concentrating around a point x₀. Given such a sequence φₙ, then ∫ φₙ g tends to g x₀ in many situations, with a whole zoo of possible assumptions on φₙ and g. This file is devoted to such results.

Main results

• tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt: If a sequence of peak functions φᵢ converges uniformly to zero away from a point x₀, and g is integrable and continuous at x₀, then ∫ φᵢ • g converges to g x₀.
• tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn: If a continuous function c realizes its maximum at a unique point x₀ in a compact set s, then the sequence of functions (c x) ^ n / ∫ (c x) ^ n is a sequence of peak functions concentrating around x₀. Therefore, ∫ (c x) ^ n * g / ∫ (c x) ^ n converges to g x₀ if g is continuous on s.

Note that there are related results about convolution with respect to peak functions in the file Analysis.Convolution, such as convolution_tendsto_right there.

theorem Set.disjoint_sdiff_inter {α : Type u_1} (s : Set α) (t : Set α) : Disjoint (s \ t) (s ∩ t)

This lemma exists for finsets, but not for sets currently. porting note: move to data.set.basic after the port.

theorem integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s : Set α} {φ : ι → α → ℝ} (hs : MeasurableSet s) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : ∀ᶠ (i : ι) in l, ∫ (x : α) in s, φ i x ∂μ = 1) (hmg : MeasureTheory.IntegrableOn g s) (hcg : ContinuousWithinAt g s x₀) : ∀ᶠ (i : ι) in l, MeasureTheory.IntegrableOn (fun (x : α) => φ i x • g x) s

If a sequence of peak functions φᵢ converges uniformly to zero away from a point x₀, and g is integrable and continuous at x₀, then φᵢ • g is eventually integrable.

theorem tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s : Set α} {φ : ι → α → ℝ} (hs : MeasurableSet s) (hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : ∀ᶠ (i : ι) in l, ∫ (x : α) in s, φ i x ∂μ = 1) (hmg : MeasureTheory.IntegrableOn g s) (h'g : g x₀ = 0) (hcg : ContinuousWithinAt g s x₀) : Filter.Tendsto (fun (i : ι) => ∫ (x : α) in s, φ i x • g x ∂μ) l (nhds 0)

If a sequence of peak functions φᵢ converges uniformly to zero away from a point x₀, and g is integrable and continuous at x₀, then ∫ φᵢ • g converges to x₀. Auxiliary lemma where one assumes additionally g x₀ = 0.

theorem tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s : Set α} {φ : ι → α → ℝ} [CompleteSpace E] (hs : MeasurableSet s) (h's : ↑↑μ s ≠ ⊤) (hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : (fun (i : ι) => ∫ (x : α) in s, φ i x ∂μ) =ᶠ[l] 1) (hmg : MeasureTheory.IntegrableOn g s) (hcg : ContinuousWithinAt g s x₀) : Filter.Tendsto (fun (i : ι) => ∫ (x : α) in s, φ i x • g x ∂μ) l (nhds (g x₀))

theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (hs : IsCompact s) (hμ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → 0 < ↑↑μ (u ∩ s)) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ s) (hmg : MeasureTheory.IntegrableOn g s) (hcg : ContinuousWithinAt g s x₀) : Filter.Tendsto (fun (n : ℕ) => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (nhds (g x₀))

If a continuous function c realizes its maximum at a unique point x₀ in a compact set s, then the sequence of functions (c x) ^ n / ∫ (c x) ^ n is a sequence of peak functions concentrating around x₀. Therefore, ∫ (c x) ^ n * g / ∫ (c x) ^ n converges to g x₀ if g is integrable on s and continuous at x₀.

Version assuming that μ gives positive mass to all neighborhoods of x₀ within s. For a less precise but more usable version, see tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn.

theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] [MeasureTheory.Measure.IsOpenPosMeasure μ] (hs : IsCompact s) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s)) (hmg : MeasureTheory.IntegrableOn g s) (hcg : ContinuousWithinAt g s x₀) : Filter.Tendsto (fun (n : ℕ) => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (nhds (g x₀))

If a continuous function c realizes its maximum at a unique point x₀ in a compact set s, then the sequence of functions (c x) ^ n / ∫ (c x) ^ n is a sequence of peak functions concentrating around x₀. Therefore, ∫ (c x) ^ n * g / ∫ (c x) ^ n converges to g x₀ if g is integrable on s and continuous at x₀.

Version assuming that μ gives positive mass to all open sets. For a less precise but more usable version, see tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn.

theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] [MeasureTheory.Measure.IsOpenPosMeasure μ] (hs : IsCompact s) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s)) (hmg : ContinuousOn g s) : Filter.Tendsto (fun (n : ℕ) => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (nhds (g x₀))

If a continuous function c realizes its maximum at a unique point x₀ in a compact set s, then the sequence of functions (c x) ^ n / ∫ (c x) ^ n is a sequence of peak functions concentrating around x₀. Therefore, ∫ (c x) ^ n * g / ∫ (c x) ^ n converges to g x₀ if g is continuous on s.