Theorem: MeasureTheory.Submartingale.sum_mul_sub
File: Mathlib/Probability/Martingale/Basic.lean

Proof steps:
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
state_after: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
tactic: have hξbdd : ∀ i, ∃ C, ∀ ω, |ξ i ω| ≤ C := fun i =>
    ⟨R, fun ω => (abs_of_nonneg (hnonneg i ω)).trans_le (hbdd i ω)⟩
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
state_after: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
tactic: have hint : ∀ m, Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ := fun m =>
    integrable_finset_sum' _ fun i _ => Integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _))
      hξ.stronglyMeasurable.aestronglyMeasurable (hξbdd _)
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
state_after: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
tactic: have hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k) := by
    intro m
    refine Finset.stronglyMeasurable_sum' _ fun i hi => ?_
    rw [Finset.mem_range] at hi
    exact (hξ.stronglyMeasurable_le hi.le).mul
      ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub
        (hf.adapted.stronglyMeasurable_le hi.le))
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
⊢ Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ
state_after: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
i : ℕ
⊢ 0 ≤ᶠ[ae μ] μ[∑ k ∈ Finset.range (i + 1), ξ k * (f (k + 1) - f k) - ∑ k ∈ Finset.range i, ξ k * (f (k + 1) - f k)|↑𝒢 i]
tactic: refine submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => ?_
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
i : ℕ
⊢ 0 ≤ᶠ[ae μ] μ[∑ k ∈ Finset.range (i + 1), ξ k * (f (k + 1) - f k) - ∑ k ∈ Finset.range i, ξ k * (f (k + 1) - f k)|↑𝒢 i]
state_after: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
i : ℕ
⊢ 0 ≤ᶠ[ae μ] μ[ξ i * (f (i + 1) - f i)|↑𝒢 i]
tactic: simp only [← Finset.sum_Ico_eq_sub _ (Nat.le_succ _), Finset.sum_apply, Pi.mul_apply,
    Pi.sub_apply, Nat.Ico_succ_singleton, Finset.sum_singleton]
state_before: Ω : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ g : ι → Ω → E
ℱ : Filtration ι m0
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k)) μ
hadp : Adapted 𝒢 fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)
i : ℕ
⊢ 0 ≤ᶠ[ae μ] μ[ξ i * (f (i + 1) - f i)|↑𝒢 i]
state_after: no goals
tactic: exact EventuallyLE.trans (EventuallyLE.mul_nonneg (eventually_of_forall (hnonneg _))
    (hf.condexp_sub_nonneg (Nat.le_succ _))) (condexp_stronglyMeasurable_mul (hξ _)
    (((hf.integrable _).sub (hf.integrable _)).bdd_mul
      hξ.stronglyMeasurable.aestronglyMeasurable (hξbdd _))
    ((hf.integrable _).sub (hf.integrable _))).symm.le