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measure_theory.integral.peak_function

Integrals against peak functions #

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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_integrable_on_of_continuous_within_at: 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_is_compact_of_continuous_on: 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 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 integrable_on_peak_smul_of_integrable_on_of_continuous_within_at {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {l : filter ι} {x₀ : α} {s : set α} {φ : ι α } (hs : measurable_set s) (hlφ : (u : set α), x₀ u l (s \ u)) (hiφ : ∀ᶠ (i : ι) in l, (x : α) in s, φ i x μ = 1) (hmg : μ) (hcg : x₀) :
∀ᶠ (i : ι) in l, measure_theory.integrable_on (λ (x : α), φ i x g x) s μ

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_integrable_on_of_continuous_within_at_aux {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {l : filter ι} {x₀ : α} {s : set α} {φ : ι α } (hs : measurable_set s) (hnφ : ∀ᶠ (i : ι) in l, (x : α), x s 0 φ i x) (hlφ : (u : set α), x₀ u l (s \ u)) (hiφ : ∀ᶠ (i : ι) in l, (x : α) in s, φ i x μ = 1) (hmg : μ) (h'g : g x₀ = 0) (hcg : x₀) :
filter.tendsto (λ (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_integrable_on_of_continuous_within_at {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {l : filter ι} {x₀ : α} {s : set α} {φ : ι α } (hs : measurable_set s) (h's : μ s ) (hnφ : ∀ᶠ (i : ι) in l, (x : α), x s 0 φ i x) (hlφ : (u : set α), x₀ u l (s \ u)) (hiφ : (λ (i : ι), (x : α) in s, φ i x μ) =ᶠ[l] 1) (hmg : μ) (hcg : x₀) :
filter.tendsto (λ (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_is_compact_of_measure_nhds_within_pos {α : Type u_1} {E : Type u_2} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {x₀ : α} {s : set α} (hs : is_compact s) (hμ : (u : set α), x₀ u 0 < μ (u s)) {c : α } (hc : s) (h'c : (y : α), y s y x₀ c y < c x₀) (hnc : (x : α), x s 0 c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ s) (hmg : μ) (hcg : x₀) :
filter.tendsto (λ (n : ), ( (x : α) in s, c x ^ n μ)⁻¹ (x : α) in s, c x ^ n g x μ) filter.at_top (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_is_compact_of_continuous_on.

theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_is_compact_of_integrable_on {α : Type u_1} {E : Type u_2} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {x₀ : α} {s : set α} (hs : is_compact s) {c : α } (hc : s) (h'c : (y : α), y s y x₀ c y < c x₀) (hnc : (x : α), x s 0 c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ closure (interior s)) (hmg : μ) (hcg : x₀) :
filter.tendsto (λ (n : ), ( (x : α) in s, c x ^ n μ)⁻¹ (x : α) in s, c x ^ n g x μ) filter.at_top (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_is_compact_of_continuous_on.

theorem tendsto_set_integral_pow_smul_of_unique_maximum_of_is_compact_of_continuous_on {α : Type u_1} {E : Type u_2} {hm : measurable_space α} {μ : measure_theory.measure α} [borel_space α] [ E] {g : α E} {x₀ : α} {s : set α} (hs : is_compact s) {c : α } (hc : s) (h'c : (y : α), y s y x₀ c y < c x₀) (hnc : (x : α), x s 0 c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ closure (interior s)) (hmg : s) :
filter.tendsto (λ (n : ), ( (x : α) in s, c x ^ n μ)⁻¹ (x : α) in s, c x ^ n g x μ) filter.at_top (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.