Theorem: MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

Proof steps:
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
this : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f n ∧ ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f n ∧ ∫⁻ a, f n a ∂μ = ∫⁻ a, g a ∂μ :=
    fun n ↦ exists_measurable_le_lintegral_eq _ _
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
this : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f n ∧ ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: choose g gmeas gf hg using this
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: let g' : ℕ → α → ℝ≥0∞ := Nat.rec (g 0) (fun n I x ↦ max (g (n+1) x) (I x))
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: have M n : Measurable (g' n) := by
    induction n with
    | zero => simp [g', gmeas 0]
    | succ n ih => exact Measurable.max (gmeas (n+1)) ih
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: have I : ∀ n x, g n x ≤ g' n x := by
    intro n x
    cases n with | zero | succ => simp [g']
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
tactic: have I' : ∀ᵐ x ∂μ, ∀ n, g' n x ≤ f n x := by
    filter_upwards [hf_mono] with x hx n
    induction n with
    | zero => simpa [g'] using gf 0 x
    | succ n ih => exact max_le (gf (n+1) x) (ih.trans (hx (Nat.le_succ n)))
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => g' i a) atTop (𝓝 (F a))
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
state_after: case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => g' i a) atTop (𝓝 (F a))
x : α
hx : Tendsto (fun i => g' i x) atTop (𝓝 (F x))
h'x : ∀ (n : ℕ), g' n x ≤ f n x
h''x : ∀ (i : ℕ), f i x ≤ F x
⊢ Tendsto (fun i => f i x) atTop (𝓝 (F x))
tactic: filter_upwards [this, I', h_bound] with x hx h'x h''x
state_before: case h
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α✝
μ✝ ν : Measure α✝
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t => Nat.rec (motive := fun x => α → ℝ≥0∞) (g 0) (fun n I x => max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i => g' i a) atTop (𝓝 (F a))
x : α
hx : Tendsto (fun i => g' i x) atTop (𝓝 (F x))
h'x : ∀ (n : ℕ), g' n x ≤ f n x
h''x : ∀ (i : ℕ), f i x ≤ F x
⊢ Tendsto (fun i => f i x) atTop (𝓝 (F x))
state_after: no goals
tactic: exact tendsto_of_tendsto_of_tendsto_of_le_of_le hx tendsto_const_nhds h'x h''x