File: Mathlib/Probability/Martingale/Basic.lean

full_name: MeasureTheory.Martingale
code: /-- A family of functions `f : ι → Ω → E` is a martingale with respect to a filtration `ℱ` if `f`
is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] =ᵐ[μ] f i`. -/
def Martingale (f : ι → Ω → E) (ℱ : Filtration ι m0) (μ : Measure Ω) : Prop 
goal: Prop

full_name: MeasureTheory.Supermartingale
code: /-- A family of integrable functions `f : ι → Ω → E` is a supermartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`μ[f j | ℱ.le i] ≤ᵐ[μ] f i`. -/
def Supermartingale [LE E] (f : ι → Ω → E) (ℱ : Filtration ι m0) (μ : Measure Ω) : Prop 
goal: Prop

full_name: MeasureTheory.Submartingale
code: /-- A family of integrable functions `f : ι → Ω → E` is a submartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`f i ≤ᵐ[μ] μ[f j | ℱ.le i]`. -/
def Submartingale [LE E] (f : ι → Ω → E) (ℱ : Filtration ι m0) (μ : Measure Ω) : Prop 
goal: Prop

full_name: MeasureTheory.martingale_const
code: theorem martingale_const (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ] (x : E) :
    Martingale (fun _ _ => x) ℱ μ 
goal: MeasureTheory.Martingale (fun x_1 x_2 => x) ℱ μ

full_name: MeasureTheory.martingale_const_fun
code: theorem martingale_const_fun [OrderBot ι] (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ]
    {f : Ω → E} (hf : StronglyMeasurable[ℱ ⊥] f) (hfint : Integrable f μ) :
    Martingale (fun _ => f) ℱ μ 
goal: MeasureTheory.Martingale (fun x => f) ℱ μ

full_name: MeasureTheory.martingale_zero
code: theorem martingale_zero (ℱ : Filtration ι m0) (μ : Measure Ω) : Martingale (0 : ι → Ω → E) ℱ μ 
goal: MeasureTheory.Martingale 0 ℱ μ

full_name: MeasureTheory.Martingale.adapted
code: protected theorem adapted (hf : Martingale f ℱ μ) : Adapted ℱ f 
goal: MeasureTheory.Adapted ℱ f

full_name: MeasureTheory.Martingale.stronglyMeasurable
code: protected theorem stronglyMeasurable (hf : Martingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
goal: MeasureTheory.StronglyMeasurable (f i)

full_name: MeasureTheory.Martingale.condexp_ae_eq
code: theorem condexp_ae_eq (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j) : μ[f j|ℱ i] =ᵐ[μ] f i 
goal: μ[f j|↑ℱ i] =ᵐ[μ] f i

full_name: MeasureTheory.Martingale.integrable
code: protected theorem integrable (hf : Martingale f ℱ μ) (i : ι) : Integrable (f i) μ 
goal: MeasureTheory.Integrable (f i) μ

full_name: MeasureTheory.Martingale.setIntegral_eq
code: theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
    {s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ 
goal: ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ

full_name: MeasureTheory.Martingale.add
code: theorem add (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f + g) ℱ μ 
goal: MeasureTheory.Martingale (f + g) ℱ μ

full_name: MeasureTheory.Martingale.neg
code: theorem neg (hf : Martingale f ℱ μ) : Martingale (-f) ℱ μ 
goal: MeasureTheory.Martingale (-f) ℱ μ

full_name: MeasureTheory.Martingale.sub
code: theorem sub (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f - g) ℱ μ 
goal: MeasureTheory.Martingale (f - g) ℱ μ

full_name: MeasureTheory.Martingale.smul
code: theorem smul (c : ℝ) (hf : Martingale f ℱ μ) : Martingale (c • f) ℱ μ 
goal: MeasureTheory.Martingale (c • f) ℱ μ

full_name: MeasureTheory.Martingale.supermartingale
code: theorem supermartingale [Preorder E] (hf : Martingale f ℱ μ) : Supermartingale f ℱ μ 
goal: MeasureTheory.Supermartingale f ℱ μ

full_name: MeasureTheory.Martingale.submartingale
code: theorem submartingale [Preorder E] (hf : Martingale f ℱ μ) : Submartingale f ℱ μ 
goal: MeasureTheory.Submartingale f ℱ μ

full_name: MeasureTheory.martingale_iff
code: theorem martingale_iff [PartialOrder E] :
    Martingale f ℱ μ ↔ Supermartingale f ℱ μ ∧ Submartingale f ℱ μ 
goal: MeasureTheory.Martingale f ℱ μ ↔ MeasureTheory.Supermartingale f ℱ μ ∧ MeasureTheory.Submartingale f ℱ μ

full_name: MeasureTheory.martingale_condexp
code: theorem martingale_condexp (f : Ω → E) (ℱ : Filtration ι m0) (μ : Measure Ω)
    [SigmaFiniteFiltration μ ℱ] : Martingale (fun i => μ[f|ℱ i]) ℱ μ 
goal: MeasureTheory.Martingale (fun i => μ[f|↑ℱ i]) ℱ μ

full_name: MeasureTheory.Supermartingale.adapted
code: protected theorem adapted [LE E] (hf : Supermartingale f ℱ μ) : Adapted ℱ f 
goal: MeasureTheory.Adapted ℱ f

full_name: MeasureTheory.Supermartingale.stronglyMeasurable
code: protected theorem stronglyMeasurable [LE E] (hf : Supermartingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
goal: MeasureTheory.StronglyMeasurable (f i)

full_name: MeasureTheory.Supermartingale.integrable
code: protected theorem integrable [LE E] (hf : Supermartingale f ℱ μ) (i : ι) : Integrable (f i) μ 
goal: MeasureTheory.Integrable (f i) μ

full_name: MeasureTheory.Supermartingale.condexp_ae_le
code: theorem condexp_ae_le [LE E] (hf : Supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
    μ[f j|ℱ i] ≤ᵐ[μ] f i 
goal: μ[f j|↑ℱ i] ≤ᵐ[μ] f i

full_name: MeasureTheory.Supermartingale.setIntegral_le
code: theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Supermartingale f ℱ μ)
    {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
    ∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ 
goal: ∫ (ω : Ω) in s, f j ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ

full_name: MeasureTheory.Supermartingale.add
code: theorem add [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Supermartingale f ℱ μ)
    (hg : Supermartingale g ℱ μ) : Supermartingale (f + g) ℱ μ 
goal: MeasureTheory.Supermartingale (f + g) ℱ μ

full_name: MeasureTheory.Supermartingale.add_martingale
code: theorem add_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Martingale g ℱ μ) : Supermartingale (f + g) ℱ μ 
goal: MeasureTheory.Supermartingale (f + g) ℱ μ

full_name: MeasureTheory.Supermartingale.neg
code: theorem neg [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Supermartingale f ℱ μ) :
    Submartingale (-f) ℱ μ 
goal: MeasureTheory.Submartingale (-f) ℱ μ

full_name: MeasureTheory.Submartingale.adapted
code: protected theorem adapted [LE E] (hf : Submartingale f ℱ μ) : Adapted ℱ f 
goal: MeasureTheory.Adapted ℱ f

full_name: MeasureTheory.Submartingale.stronglyMeasurable
code: protected theorem stronglyMeasurable [LE E] (hf : Submartingale f ℱ μ) (i : ι) :
    StronglyMeasurable[ℱ i] (f i) 
goal: MeasureTheory.StronglyMeasurable (f i)

full_name: MeasureTheory.Submartingale.integrable
code: protected theorem integrable [LE E] (hf : Submartingale f ℱ μ) (i : ι) : Integrable (f i) μ 
goal: MeasureTheory.Integrable (f i) μ

full_name: MeasureTheory.Submartingale.ae_le_condexp
code: theorem ae_le_condexp [LE E] (hf : Submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
    f i ≤ᵐ[μ] μ[f j|ℱ i] 
goal: f i ≤ᵐ[μ] μ[f j|↑ℱ i]

full_name: MeasureTheory.Submartingale.add
code: theorem add [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Submartingale g ℱ μ) : Submartingale (f + g) ℱ μ 
goal: MeasureTheory.Submartingale (f + g) ℱ μ

full_name: MeasureTheory.Submartingale.add_martingale
code: theorem add_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Martingale g ℱ μ) : Submartingale (f + g) ℱ μ 
goal: MeasureTheory.Submartingale (f + g) ℱ μ

full_name: MeasureTheory.Submartingale.neg
code: theorem neg [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ) :
    Supermartingale (-f) ℱ μ 
goal: MeasureTheory.Supermartingale (-f) ℱ μ

full_name: MeasureTheory.Submartingale.setIntegral_le
code: /-- The converse of this lemma is `MeasureTheory.submartingale_of_setIntegral_le`. -/
theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ)
    {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
    ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ 
goal: ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ

full_name: MeasureTheory.Submartingale.sub_supermartingale
code: theorem sub_supermartingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Submartingale f ℱ μ) (hg : Supermartingale g ℱ μ) : Submartingale (f - g) ℱ μ 
goal: MeasureTheory.Submartingale (f - g) ℱ μ

full_name: MeasureTheory.Submartingale.sub_martingale
code: theorem sub_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)] (hf : Submartingale f ℱ μ)
    (hg : Martingale g ℱ μ) : Submartingale (f - g) ℱ μ 
goal: MeasureTheory.Submartingale (f - g) ℱ μ

full_name: MeasureTheory.Submartingale.sup
code: protected theorem sup {f g : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) (hg : Submartingale g ℱ μ) :
    Submartingale (f ⊔ g) ℱ μ 
goal: MeasureTheory.Submartingale (f ⊔ g) ℱ μ

full_name: MeasureTheory.Submartingale.pos
code: protected theorem pos {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) : Submartingale (f⁺) ℱ μ 
goal: MeasureTheory.Submartingale f⁺ ℱ μ

full_name: MeasureTheory.submartingale_of_setIntegral_le
code: theorem submartingale_of_setIntegral_le [IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : Adapted ℱ f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i j : ι,
      i ≤ j → ∀ s : Set Ω, MeasurableSet[ℱ i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ) :
    Submartingale f ℱ μ 
goal: MeasureTheory.Submartingale f ℱ μ

full_name: MeasureTheory.submartingale_of_condexp_sub_nonneg
code: theorem submartingale_of_condexp_sub_nonneg [IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : Adapted ℱ f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|ℱ i]) :
    Submartingale f ℱ μ 
goal: MeasureTheory.Submartingale f ℱ μ

full_name: MeasureTheory.Submartingale.condexp_sub_nonneg
code: theorem Submartingale.condexp_sub_nonneg {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) {i j : ι}
    (hij : i ≤ j) : 0 ≤ᵐ[μ] μ[f j - f i|ℱ i] 
goal: 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i]

full_name: MeasureTheory.submartingale_iff_condexp_sub_nonneg
code: theorem submartingale_iff_condexp_sub_nonneg [IsFiniteMeasure μ] {f : ι → Ω → ℝ} :
    Submartingale f ℱ μ ↔
      Adapted ℱ f ∧ (∀ i, Integrable (f i) μ) ∧ ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|ℱ i] 
goal: MeasureTheory.Submartingale f ℱ μ ↔
  MeasureTheory.Adapted ℱ f ∧
    (∀ (i : ι), MeasureTheory.Integrable (f i) μ) ∧ ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i]

full_name: MeasureTheory.Supermartingale.sub_submartingale
code: theorem sub_submartingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Submartingale g ℱ μ) : Supermartingale (f - g) ℱ μ 
goal: MeasureTheory.Supermartingale (f - g) ℱ μ

full_name: MeasureTheory.Supermartingale.sub_martingale
code: theorem sub_martingale [Preorder E] [CovariantClass E E (· + ·) (· ≤ ·)]
    (hf : Supermartingale f ℱ μ) (hg : Martingale g ℱ μ) : Supermartingale (f - g) ℱ μ 
goal: MeasureTheory.Supermartingale (f - g) ℱ μ

full_name: MeasureTheory.Supermartingale.smul_nonneg
code: theorem smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : Supermartingale f ℱ μ) :
    Supermartingale (c • f) ℱ μ 
goal: MeasureTheory.Supermartingale (c • f) ℱ μ

full_name: MeasureTheory.Supermartingale.smul_nonpos
code: theorem smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : Supermartingale f ℱ μ) :
    Submartingale (c • f) ℱ μ 
goal: MeasureTheory.Submartingale (c • f) ℱ μ

full_name: MeasureTheory.Submartingale.smul_nonneg
code: theorem smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : Submartingale f ℱ μ) :
    Submartingale (c • f) ℱ μ 
goal: MeasureTheory.Submartingale (c • f) ℱ μ

full_name: MeasureTheory.Submartingale.smul_nonpos
code: theorem smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : Submartingale f ℱ μ) :
    Supermartingale (c • f) ℱ μ 
goal: MeasureTheory.Supermartingale (c • f) ℱ μ

full_name: MeasureTheory.submartingale_of_setIntegral_le_succ
code: theorem submartingale_of_setIntegral_le_succ [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f (i + 1) ω ∂μ) :
    Submartingale f 𝒢 μ 
goal: MeasureTheory.Submartingale f 𝒢 μ

full_name: MeasureTheory.supermartingale_of_setIntegral_succ_le
code: theorem supermartingale_of_setIntegral_succ_le [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f (i + 1) ω ∂μ ≤ ∫ ω in s, f i ω ∂μ) :
    Supermartingale f 𝒢 μ 
goal: MeasureTheory.Supermartingale f 𝒢 μ

full_name: MeasureTheory.martingale_of_setIntegral_eq_succ
code: theorem martingale_of_setIntegral_eq_succ [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f i ω ∂μ = ∫ ω in s, f (i + 1) ω ∂μ) :
    Martingale f 𝒢 μ 
goal: MeasureTheory.Martingale f 𝒢 μ

full_name: MeasureTheory.submartingale_nat
code: theorem submartingale_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i, f i ≤ᵐ[μ] μ[f (i + 1)|𝒢 i]) :
    Submartingale f 𝒢 μ 
goal: MeasureTheory.Submartingale f 𝒢 μ

full_name: MeasureTheory.supermartingale_nat
code: theorem supermartingale_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i, μ[f (i + 1)|𝒢 i] ≤ᵐ[μ] f i) :
    Supermartingale f 𝒢 μ 
goal: MeasureTheory.Supermartingale f 𝒢 μ

full_name: MeasureTheory.martingale_nat
code: theorem martingale_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ} (hadp : Adapted 𝒢 f)
    (hint : ∀ i, Integrable (f i) μ) (hf : ∀ i, f i =ᵐ[μ] μ[f (i + 1)|𝒢 i]) : Martingale f 𝒢 μ 
goal: MeasureTheory.Martingale f 𝒢 μ

full_name: MeasureTheory.submartingale_of_condexp_sub_nonneg_nat
code: theorem submartingale_of_condexp_sub_nonneg_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, 0 ≤ᵐ[μ] μ[f (i + 1) - f i|𝒢 i]) : Submartingale f 𝒢 μ 
goal: MeasureTheory.Submartingale f 𝒢 μ

full_name: MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat
code: theorem supermartingale_of_condexp_sub_nonneg_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, 0 ≤ᵐ[μ] μ[f i - f (i + 1)|𝒢 i]) : Supermartingale f 𝒢 μ 
goal: MeasureTheory.Supermartingale f 𝒢 μ

full_name: MeasureTheory.martingale_of_condexp_sub_eq_zero_nat
code: theorem martingale_of_condexp_sub_eq_zero_nat [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
    (hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
    (hf : ∀ i, μ[f (i + 1) - f i|𝒢 i] =ᵐ[μ] 0) : Martingale f 𝒢 μ 
goal: MeasureTheory.Martingale f 𝒢 μ

full_name: MeasureTheory.Submartingale.zero_le_of_predictable
code: /-- A predictable submartingale is a.e. greater equal than its initial state. -/
theorem Submartingale.zero_le_of_predictable [Preorder E] [SigmaFiniteFiltration μ 𝒢]
    {f : ℕ → Ω → E} (hfmgle : Submartingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1)) (n : ℕ) :
    f 0 ≤ᵐ[μ] f n 
goal: f 0 ≤ᵐ[μ] f n

full_name: MeasureTheory.Supermartingale.le_zero_of_predictable
code: /-- A predictable supermartingale is a.e. less equal than its initial state. -/
theorem Supermartingale.le_zero_of_predictable [Preorder E] [SigmaFiniteFiltration μ 𝒢]
    {f : ℕ → Ω → E} (hfmgle : Supermartingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1))
    (n : ℕ) : f n ≤ᵐ[μ] f 0 
goal: f n ≤ᵐ[μ] f 0

full_name: MeasureTheory.Martingale.eq_zero_of_predictable
code: /-- A predictable martingale is a.e. equal to its initial state. -/
theorem Martingale.eq_zero_of_predictable [SigmaFiniteFiltration μ 𝒢] {f : ℕ → Ω → E}
    (hfmgle : Martingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1)) (n : ℕ) : f n =ᵐ[μ] f 0 
goal: f n =ᵐ[μ] f 0

full_name: MeasureTheory.Submartingale.integrable_stoppedValue
code: protected theorem integrable_stoppedValue [LE E] {f : ℕ → Ω → E} (hf : Submartingale f 𝒢 μ)
    {τ : Ω → ℕ} (hτ : IsStoppingTime 𝒢 τ) {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) :
    Integrable (stoppedValue f τ) μ 
goal: MeasureTheory.Integrable (MeasureTheory.stoppedValue f τ) μ

full_name: MeasureTheory.Submartingale.sum_mul_sub
code: theorem Submartingale.sum_mul_sub [IsFiniteMeasure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ}
    (hf : Submartingale f 𝒢 μ) (hξ : Adapted 𝒢 ξ) (hbdd : ∀ n ω, ξ n ω ≤ R)
    (hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
    Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ 
goal: MeasureTheory.Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ

full_name: MeasureTheory.Submartingale.sum_mul_sub'
code: /-- Given a discrete submartingale `f` and a predictable process `ξ` (i.e. `ξ (n + 1)` is adapted)
the process defined by `fun n => ∑ k ∈ Finset.range n, ξ (k + 1) * (f (k + 1) - f k)` is also a
submartingale. -/
theorem Submartingale.sum_mul_sub' [IsFiniteMeasure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ}
    (hf : Submartingale f 𝒢 μ) (hξ : Adapted 𝒢 fun n => ξ (n + 1)) (hbdd : ∀ n ω, ξ n ω ≤ R)
    (hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
    Submartingale (fun n => ∑ k ∈ Finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ 
goal: MeasureTheory.Submartingale (fun n => ∑ k ∈ Finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ