ProbabilityTheory.HasCondSubexponentialMGF
From the authors
A random variable X has a conditionally sub-exponential moment-generating function
with parameters (V, b) with respect to a sigma-algebra m and a measure μ if for all t
with b * |t| ≤ 1, exp (t * X) is μ-integrable and the moment-generating function of X
conditioned on m is almost surely bounded by exp (V * t ^ 2 / 2).
The actual definition uses Kernel.HasSubexponentialMGF: HasCondSubexponentialMGF is defined
as sub-exponential with respect to the conditional expectation kernel for m and the restriction
of μ to the sigma-algebra m.
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Ω : Type u_1m : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.mΩ : MeasurableSpace ΩStandardBorelSpace ΩA standard Borel space is a measurable space arising as the Borel sets of some Polish topology.
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X : Ω → ℝ -
V : ℝ -
b : ℝ -
μ : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsFiniteMeasure μA measureμis called finite ifμ univ < ∞.
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hm : m ≤ mΩ
PropKernel.HasSubexponentialMGF X V b (condExpKernel μ m) (μ.trim hm)MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
StandardBorelSpace : (α : Type u_1) → [MeasurableSpace α] → PropA standard Borel space is a measurable space arising as the Borel sets of some Polish topology. This is useful in situations where a space has no natural topology or the natural topology in a space is non-Polish. To endow a standard Borel space `α` with a compatible Polish topology, use `letI := upgradeStandardBorel α`. One can then use `eq_borel_upgradeStandardBorel α` to rewrite the `MeasurableSpace α` instance to `borel α t`, where `t` is the new topology.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called finite if `μ univ < ∞`.
MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version is written `μ.real s`.
LE.le : {α : Type u} → [self : LE α] → α → α → PropThe less-equal relation: `x ≤ y` Conventions for notations in identifiers: * The recommended spelling of `≤` in identifiers is `le`.
ProbabilityTheory.Kernel.HasSubexponentialMGF : {Ω : Type u_1} →
{Ω' : Type u_2} →
{mΩ : MeasurableSpace Ω} →
{mΩ' : MeasurableSpace Ω'} →
(Ω → ℝ) →
ℝ →
ℝ →
ProbabilityTheory.Kernel Ω' Ω →
autoParam (MeasureTheory.Measure Ω') ProbabilityTheory.Kernel.HasSubexponentialMGF._auto_1 → PropA random variable `X` has a sub-exponential moment-generating function with parameters `(V, b)` with respect to a kernel `κ` and a measure `ν` if for every `t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable with respect to `κ ∘ₘ ν` and, for `ν`-almost all `ω'`, the moment-generating function of `X` with respect to `κ ω'` is bounded by `exp (V * t ^ 2 / 2)`. For `b = 0` this is `Kernel.HasSubgaussianMGF X V κ ν`.Go to its page
ProbabilityTheory.condExpKernel : {Ω : Type u_3} →
[mΩ : MeasurableSpace Ω] →
[StandardBorelSpace Ω] →
(μ : MeasureTheory.Measure Ω) →
[MeasureTheory.IsFiniteMeasure μ] → (m : MeasurableSpace Ω) → ProbabilityTheory.Kernel Ω ΩKernel associated with the conditional expectation with respect to a σ-algebra. It satisfies `μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condExpKernel μ m ω)`. It is defined as the conditional distribution of the identity given the identity, where the second identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` with σ-algebra `m ⊓ mΩ`. We use `m ⊓ mΩ` instead of `m` to ensure that it is a sub-σ-algebra of `mΩ`. We then use `Kernel.comap` to get a kernel from `m` to `mΩ` instead of from `m ⊓ mΩ` to `mΩ`.
MeasureTheory.Measure.trim : {α : Type u_1} → {m m0 : MeasurableSpace α} → MeasureTheory.Measure α → m ≤ m0 → MeasureTheory.Measure αRestriction of a measure to a sub-σ-algebra. It is common to see a measure `μ` on a measurable space structure `m0` as being also a measure on any `m ≤ m0`. Since measures in mathlib have to be trimmed to the measurable space, `μ` itself cannot be a measure on `m`, hence the definition of `μ.trim hm`. This notion is related to `OuterMeasure.trim`, see the lemma `toOuterMeasure_trim_eq_trim_toOuterMeasure`.
Code
def HasCondSubexponentialMGF (X : Ω → ℝ) (V b : ℝ)
(μ : Measure Ω := by volume_tac) [IsFiniteMeasure μ] : Prop :=
Kernel.HasSubexponentialMGF X V b (condExpKernel μ m) (μ.trim hm)Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 1 project declarations, 65 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.