ProbabilityTheory.HasSubexponentialMGF.trim
Lemma
No docstring.
Types
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Ω : Type u_1m : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.mΩ : MeasurableSpace Ω
Given
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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. -
X : Ω → ℝ -
V : ℝ -
b : ℝ
Assuming
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hm : m ≤ mΩ -
hXm : Measurable XA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable. -
hX : HasSubexponentialMGF X V b μXhas a sub-exponential moment generating function with parameters(V, b): for everytwithb * |t| ≤ 1,exp (t * X)is integrable andmgf X μ t ≤ exp (V * t ^ 2 / 2).
Then
HasSubexponentialMGF X V b (μ.trim hm)MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
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`.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
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`.
Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
ProbabilityTheory.HasSubexponentialMGF : {Ω : Type u_1} →
{mΩ : MeasurableSpace Ω} →
(Ω → ℝ) → ℝ → ℝ → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasSubexponentialMGF._auto_1 → Prop`X` has a sub-exponential moment generating function with parameters `(V, b)`: for every `t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable and `mgf X μ t ≤ exp (V * t ^ 2 / 2)`. For `b = 0` this is `HasSubgaussianMGF X V μ`. This is equivalent to `Kernel.HasSubexponentialMGF X V b (Kernel.const Unit μ) (Measure.dirac ())`, as proved in `hasSubexponentialMGF_iff_kernel`.Go to its page
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
lemma trim (hm : m ≤ mΩ) (hXm : Measurable[m] X) (hX : HasSubexponentialMGF X V b μ) :
HasSubexponentialMGF X V b (μ.trim hm) where
integrable_exp_mul t htProof
by
refine (hX.integrable_exp_mul t ht).trim hm ?_
exact Measurable.stronglyMeasurable <| by fun_prop
mgf_le t ht := by
rw [mgf, ← integral_trim]
· exact hX.mgf_le t ht
· exact Measurable.stronglyMeasurable <| by fun_propMeaning 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, 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.