ProbabilityTheory.HasSubexponentialMGF.aestronglyMeasurable
Lemma
No docstring.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
Given
-
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
Then
MeasureTheory.AEStronglyMeasurable X μAEStronglyMeasurable with respect to a measure μ if it is almost everywhere equal to the limit of a sequence of simple functions.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.
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.AEStronglyMeasurable : {α : Type u_1} →
{β : Type u_2} →
[TopologicalSpace β] →
[m : MeasurableSpace α] →
{m₀ : MeasurableSpace α} →
(α → β) → autoParam (MeasureTheory.Measure α) MeasureTheory.AEStronglyMeasurable._auto_1 → PropA function is `AEStronglyMeasurable` with respect to a measure `μ` if it is almost everywhere equal to the limit of a sequence of simple functions. One can specify the sigma-algebra according to which simple functions are taken using the `AEStronglyMeasurable[m]` notation in the `MeasureTheory` scope.
Code
lemma aestronglyMeasurable (h : HasSubexponentialMGF X V b μ) : AEStronglyMeasurable X μ
Proof
by rw [hasSubexponentialMGF_iff_kernel] at h simpa using h.aestronglyMeasurable
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, 53 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.