`ProbabilityTheory.HasSubexponentialMGF.memLp`🔗

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
p : NNReal

Assuming

h : HasSubexponentialMGF X V b μX 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).

Then

MeasureTheory.MemLp X (↑p) μ

The property that f : α → E is a.e. strongly measurable and (∫ ‖f a‖ ^ p ∂μ) ^ (1/p) is finite if p < ∞, or essSup ‖f‖ < ∞ if p = ∞.

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A 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 : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

NNReal : Type

Nonnegative real numbers, denoted as `ℝ≥0` within the NNReal namespace

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.MemLp : {α : Type u_1} →
  {ε : Type u_2} →
    {m0 : MeasurableSpace α} →
      [ENorm ε] →
        [TopologicalSpace ε] →
          (α → ε) → ENNReal → autoParam (MeasureTheory.Measure α) MeasureTheory.MemLp._auto_1 → Prop

The property that `f : α → E` is a.e. strongly measurable and `(∫ ‖f a‖ ^ p ∂μ) ^ (1/p)`
is finite if `p < ∞`, or `essSup ‖f‖ < ∞` if `p = ∞`.

Code

lemma memLp (h : HasSubexponentialMGF X V b μ) (p : ℝ≥0) : MemLp X p μ

Proof

memLp_of_mem_interior_integrableExpSet h.zero_mem_interior_integrableExpSet p

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, 57 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.HasSubexponentialMGF.memLp🔗

`ProbabilityTheory.HasSubexponentialMGF.memLp`🔗