`ProbabilityTheory.HasSubgaussianMGF.measure_sum_le_sum_le`🔗

Lemma

No docstring.

theorem

ProbabilityTheory.HasSubgaussianMGF.measure_sum_le_sum_le.{u_1, u_2,
    u_3}
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
  {ι : Type u_2} {ι' : Type u_3} {X : ι → Ω → ℝ} {cX : ι → NNReal}
  {s : Finset ι} {Y : ι' → Ω → ℝ} {cY : ι' → NNReal} {t : Finset ι'}
  [MeasureTheory.IsFiniteMeasure μ] (hX_indep : iIndepFun X μ)
  (hY_indep : iIndepFun Y μ)
  (hX_subG :
    ∀ i ∈ s,
      HasSubgaussianMGF (fun ω => X i ω - ∫ (x : Ω), X i x ∂μ) (cX i) μ)
  (hY_subG :
    ∀ j ∈ t,
      HasSubgaussianMGF (fun ω => Y j ω - ∫ (x : Ω), Y j x ∂μ) (cY j) μ)
  (h_indep_sum :
    IndepFun (fun ω => ∑ i ∈ s, X i ω) (fun ω => ∑ j ∈ t, Y j ω) μ)
  (h_le : ∑ j ∈ t, ∫ (x : Ω), Y j x ∂μ ≤ ∑ i ∈ s, ∫ (x : Ω), X i x ∂μ) :
  MeasureTheory.Measure.real μ {ω | ∑ i ∈ s, X i ω ≤ ∑ j ∈ t, Y j ω} ≤
    Real.exp
      (-(∑ j ∈ t, ∫ (x : Ω), Y j x ∂μ - ∑ i ∈ s, ∫ (x : Ω), X i x ∂μ) ^
            2 /
        (2 * (↑(∑ i ∈ s, cX i) + ↑(∑ j ∈ t, cY j))))
ProbabilityTheory.HasSubgaussianMGF.measure_sum_le_sum_le.{u_1,
    u_2, u_3}
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure Ω}
  {ι : Type u_2} {ι' : Type u_3}
  {X : ι → Ω → ℝ} {cX : ι → NNReal}
  {s : Finset ι} {Y : ι' → Ω → ℝ}
  {cY : ι' → NNReal} {t : Finset ι'}
  [MeasureTheory.IsFiniteMeasure μ]
  (hX_indep : iIndepFun X μ)
  (hY_indep : iIndepFun Y μ)
  (hX_subG :
    ∀ i ∈ s,
      HasSubgaussianMGF
        (fun ω =>
          X i ω - ∫ (x : Ω), X i x ∂μ)
        (cX i) μ)
  (hY_subG :
    ∀ j ∈ t,
      HasSubgaussianMGF
        (fun ω =>
          Y j ω - ∫ (x : Ω), Y j x ∂μ)
        (cY j) μ)
  (h_indep_sum :
    IndepFun (fun ω => ∑ i ∈ s, X i ω)
      (fun ω => ∑ j ∈ t, Y j ω) μ)
  (h_le :
    ∑ j ∈ t, ∫ (x : Ω), Y j x ∂μ ≤
      ∑ i ∈ s, ∫ (x : Ω), X i x ∂μ) :
  MeasureTheory.Measure.real μ
      {ω |
        ∑ i ∈ s, X i ω ≤ ∑ j ∈ t, Y j ω} ≤
    Real.exp
      (-(∑ j ∈ t, ∫ (x : Ω), Y j x ∂μ -
              ∑ i ∈ s,
                ∫ (x : Ω), X i x ∂μ) ^
            2 /
        (2 *
          (↑(∑ i ∈ s, cX i) +
            ↑(∑ j ∈ t, cY j))))

Code

lemma measure_sum_le_sum_le [IsFiniteMeasure μ]
    (hX_indep : iIndepFun X μ) (hY_indep : iIndepFun Y μ)
    (hX_subG : ∀ i ∈ s, HasSubgaussianMGF (fun ω ↦ X i ω - μ[X i]) (cX i) μ)
    (hY_subG : ∀ j ∈ t, HasSubgaussianMGF (fun ω ↦ Y j ω - μ[Y j]) (cY j) μ)
    (h_indep_sum : IndepFun (fun ω ↦ ∑ i ∈ s, X i ω) (fun ω ↦ ∑ j ∈ t, Y j ω) μ)
    (h_le : ∑ j ∈ t, μ[Y j] ≤ ∑ i ∈ s, μ[X i]) :
    μ.real {ω | ∑ i ∈ s, X i ω ≤ ∑ j ∈ t, Y j ω}
      ≤ exp (- (∑ j ∈ t, μ[Y j] - ∑ i ∈ s, μ[X i]) ^ 2
        / (2 * (∑ i ∈ s, cX i + ∑ j ∈ t, cY j)))

Proof

by
  have hX_int i (his : i ∈ s) : Integrable (X i) μ := by
    have h_int := (hX_subG i his).integrable
    simp_rw [sub_eq_add_neg, integrable_add_const_iff] at h_int
    exact h_int
  have hY_int j (his : j ∈ t) : Integrable (Y j) μ := by
    have h_int := (hY_subG j his).integrable
    simp_rw [sub_eq_add_neg, integrable_add_const_iff] at h_int
    exact h_int
  refine (measureReal_le_le_exp
    (cX := ∑ i ∈ s, cX i) (cY := ∑ j ∈ t, cY j) ?_ ?_ h_indep_sum ?_).trans_eq ?_
  · suffices HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, (X i ω - μ[X i])) (∑ i ∈ s, cX i) μ by
      convert this
      rw [integral_finsetSum _ hX_int, Finset.sum_sub_distrib]
    refine sum_of_iIndepFun ?_ hX_subG
    exact hX_indep.comp (g := fun i x ↦ x - μ[X i]) (by fun_prop)
  · suffices HasSubgaussianMGF (fun ω ↦ ∑ j ∈ t, (Y j ω - μ[Y j])) (∑ j ∈ t, cY j) μ by
      convert this
      rw [integral_finsetSum _ hY_int, Finset.sum_sub_distrib]
    refine sum_of_iIndepFun ?_ hY_subG
    exact hY_indep.comp (g := fun i x ↦ x - μ[Y i]) (by fun_prop)
  · rwa [integral_finsetSum _ hX_int, integral_finsetSum _ hY_int]
  · congr
    · rw [integral_finsetSum _ hY_int]
    · rw [integral_finsetSum _ hX_int]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 2th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 56 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

ProbabilityTheory.HasSubgaussianMGF.measure_sum_le_sum_le🔗

`ProbabilityTheory.HasSubgaussianMGF.measure_sum_le_sum_le`🔗