`ProbabilityTheory.HasSubgaussianMGF.measure_le_le`🔗

Lemma

Chernoff bound on the left tail of a sub-Gaussian random variable.

theorem

ProbabilityTheory.HasSubgaussianMGF.measure_le_le.{u_1} {Ω : Type u_1}
  {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ}
  {c : NNReal} (h : HasSubgaussianMGF X c μ) {ε : ℝ} (hε : 0 ≤ ε) :
  MeasureTheory.Measure.real μ {ω | X ω ≤ -ε} ≤
    Real.exp (-ε ^ 2 / (2 * ↑c))
ProbabilityTheory.HasSubgaussianMGF.measure_le_le.{u_1}
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure Ω}
  {X : Ω → ℝ} {c : NNReal}
  (h : HasSubgaussianMGF X c μ) {ε : ℝ}
  (hε : 0 ≤ ε) :
  MeasureTheory.Measure.real μ
      {ω | X ω ≤ -ε} ≤
    Real.exp (-ε ^ 2 / (2 * ↑c))

Code

lemma measure_le_le (h : HasSubgaussianMGF X c μ) {ε : ℝ} (hε : 0 ≤ ε) :
    μ.real {ω | X ω ≤ -ε} ≤ exp (-ε ^ 2 / (2 * c))

Proof

by
  simp_rw [le_neg (b := ε), ← Pi.neg_apply]
  exact h.neg.measure_ge_le hε

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, 34 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_le_le🔗

`ProbabilityTheory.HasSubgaussianMGF.measure_le_le`🔗