This commit is contained in:
Stefan Kebekus 2024-08-14 14:12:57 +02:00
parent 38179d24c0
commit b2de8dbc44
1 changed files with 159 additions and 19 deletions

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@ -8,12 +8,142 @@ open Real Filter MeasureTheory intervalIntegral
-- The following theorem was suggested by Gareth Ma on Zulip -- The following theorem was suggested by Gareth Ma on Zulip
lemma logsinBound : ∀ x ∈ (Set.Icc 0 1), ‖(log ∘ sin) x‖ ≤ ‖log ((π / 2)⁻¹ * x)‖ := by
intro x hx
by_cases h'x : x = 0
· rw [h'x]; simp
-- Now handle the case where x ≠ 0
have l₀ : log ((π / 2)⁻¹ * x) ≤ 0 := by
-- log_nonpos (Set.mem_Icc.1 hx).1 (Set.mem_Icc.1 hx).2
sorry
have l₁ : 0 ≤ sin x := by
apply sin_nonneg_of_nonneg_of_le_pi (Set.mem_Icc.1 hx).1
trans (1 : )
exact (Set.mem_Icc.1 hx).2
trans π / 2
exact one_le_pi_div_two
norm_num [pi_nonneg]
have l₂ : log (sin x) ≤ 0 := log_nonpos l₁ (sin_le_one x)
simp only [norm_eq_abs, Function.comp_apply]
rw [abs_eq_neg_self.2 l₀]
rw [abs_eq_neg_self.2 l₂]
simp only [neg_le_neg_iff, ge_iff_le]
have l₃ : x ∈ (Set.Ioi 0) := by
simp
exact lt_of_le_of_ne (Set.mem_Icc.1 hx).1 ( fun a => h'x (id (Eq.symm a)) )
have l₄ : sin x ∈ (Set.Ioi 0) := by
have t₁ : 0 ∈ Set.Icc (-(π / 2)) (π / 2) := by
simp
apply div_nonneg pi_nonneg zero_le_two
have t₂ : x ∈ Set.Icc (-(π / 2)) (π / 2) := by
simp
constructor
· trans 0
simp
apply div_nonneg pi_nonneg zero_le_two
exact (Set.mem_Icc.1 hx).1
· trans (1 : )
exact (Set.mem_Icc.1 hx).2
exact one_le_pi_div_two
let A := Real.strictMonoOn_sin t₁ t₂ l₃
simp at A
simpa
have l₅ : 0 < (π / 2)⁻¹ * x := by
apply mul_pos
apply inv_pos.2
apply div_pos pi_pos zero_lt_two
exact l₃
have : ∀ x ∈ (Set.Icc 0 (π / 2)), (π / 2)⁻¹ * x ≤ sin x := by
intro x hx
have i₀ : 0 ∈ Set.Icc 0 π :=
Set.left_mem_Icc.mpr pi_nonneg
have i₁ : π / 2 ∈ Set.Icc 0 π :=
Set.mem_Icc.mpr ⟨div_nonneg pi_nonneg zero_le_two, half_le_self pi_nonneg⟩
have i₂ : 0 ≤ 1 - (π / 2)⁻¹ * x := by
rw [sub_nonneg]
calc (π / 2)⁻¹ * x
_ ≤ (π / 2)⁻¹ * (π / 2) := by
apply mul_le_mul_of_nonneg_left
exact (Set.mem_Icc.1 hx).2
apply inv_nonneg.mpr (div_nonneg pi_nonneg zero_le_two)
_ = 1 := by
apply inv_mul_cancel
apply div_ne_zero_iff.mpr
constructor
· exact pi_ne_zero
· exact Ne.symm (NeZero.ne' 2)
have i₃ : 0 ≤ (π / 2)⁻¹ * x := by
apply mul_nonneg
apply inv_nonneg.2
apply div_nonneg
exact pi_nonneg
exact zero_le_two
exact (Set.mem_Icc.1 hx).1
have i₄ : 1 - (π / 2)⁻¹ * x + (π / 2)⁻¹ * x = 1 := by ring
let B := strictConcaveOn_sin_Icc.concaveOn.2 i₀ i₁ i₂ i₃ i₄
simp [Real.sin_pi_div_two] at B
rw [(by ring_nf; rw [mul_inv_cancel pi_ne_zero, one_mul] : 2 / π * x * (π / 2) = x)] at B
simpa
apply log_le_log l₅
apply this
apply Set.mem_Icc.mpr
constructor
· exact le_of_lt l₃
· trans 1
exact (Set.mem_Icc.1 hx).2
exact one_le_pi_div_two
example : IntervalIntegrable (log ∘ sin) volume 0 1 := by example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
have int_log : IntervalIntegrable log volume 0 1 := by sorry have int_log : IntervalIntegrable (fun x ↦ ‖log x‖) volume 0 1 := by
apply IntervalIntegrable.norm
rw [← neg_neg log]
apply IntervalIntegrable.neg
apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
· intro x hx
norm_num at hx
convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
norm_num
· intro x hx
norm_num at hx
rw [Pi.neg_apply, Left.nonneg_neg_iff]
exact (log_nonpos_iff hx.left).mpr hx.right.le
apply IntervalIntegrable.mono_fun' (g := log)
have int_log : IntervalIntegrable (fun x ↦ ‖log ((π / 2)⁻¹ * x)‖) volume 0 1 := by
have A := IntervalIntegrable.comp_mul_right int_log (π / 2)⁻¹
simp only [norm_eq_abs] at A
conv =>
arg 1
intro x
rw [mul_comm]
simp only [norm_eq_abs]
apply IntervalIntegrable.mono A
simp
trans Set.Icc 0 (π / 2)
exact Set.Icc_subset_Icc (Preorder.le_refl 0) one_le_pi_div_two
exact Set.Icc_subset_uIcc
exact Preorder.le_refl volume
apply IntervalIntegrable.mono_fun' (g := fun x ↦ ‖log ((π / 2)⁻¹ * x)‖)
exact int_log exact int_log
-- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1)) -- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
@ -24,32 +154,42 @@ example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
intro x hx intro x hx
by_contra contra by_contra contra
simp at contra simp at contra
rw [contra] at hx rw [contra, Set.left_mem_uIoc] at hx
rw [Set.left_mem_uIoc] at hx
linarith linarith
exact continuousOn_sin exact continuousOn_sin
--
-- Set.MapsTo sin (Ι 0 1) (Ι 0 1)
rw [Set.uIoc_of_le (zero_le_one' )] rw [Set.uIoc_of_le (zero_le_one' )]
exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩ exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
--
exact measurableSet_uIoc
--
have : ∀ x ∈ (Ι 0 1), ‖(log ∘ sin) x‖ ≤ log x := by sorry -- MeasurableSet (Ι 0 1)
exact measurableSet_uIoc
-- (fun x => ‖(log ∘ sin) x‖) ≤ᶠ[ae (volume.restrict (Ι 0 1))] ‖log‖
dsimp [EventuallyLE] dsimp [EventuallyLE]
rw [MeasureTheory.ae_restrict_iff] rw [MeasureTheory.ae_restrict_iff]
apply MeasureTheory.ae_of_all apply MeasureTheory.ae_of_all
exact this intro x hx
have : x ∈ Set.Icc 0 1 := by
simp
simp at hx
constructor
· exact le_of_lt hx.1
· exact hx.2
let A := logsinBound x this
simp only [Function.comp_apply, norm_eq_abs] at A
exact A
--intro x apply measurableSet_le
rw [MeasureTheory.ae_iff] apply Measurable.comp'
simp exact continuous_abs.measurable
exact Measurable.comp' measurable_log continuous_sin.measurable
rw [MeasureTheory.ae_iff] -- Measurable fun a => |log ((π / 2)⁻¹ * a)|
simp apply Measurable.comp'
exact continuous_abs.measurable
apply Measurable.comp'
sorry exact measurable_log
exact measurable_const_mul (π / 2)⁻¹
theorem logInt theorem logInt