ProbabilityTheory.Kernel.HasSubexponentialMGF.memLp_exp_mul
Lemma
No docstring.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
Ω' : Type u_2mΩ' : MeasurableSpace Ω'
Given
-
ν : MeasureTheory.Measure Ω'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. -
κ : Kernel Ω' ΩA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β. -
X : Ω → ℝ -
V : ℝ -
b : ℝ -
t : ℝ -
p : NNReal
Assuming
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h : HasSubexponentialMGF X V b κ νA random variableXhas a sub-exponential moment-generating function with parameters(V, b)with respect to a kernelκand a measureνif for everytwithb * |t| ≤ 1,exp (t * X)is int… -
ht : b * |↑p * t| ≤ 1
Then
MeasureTheory.MemLp (fun ω => Real.exp (t * X ω)) (↑p) (ν.bind ⇑κ)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_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`.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
NNReal : TypeNonnegative real numbers, denoted as `ℝ≥0` within the NNReal namespace
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
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`.
HMul.hMul : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ`a * b` computes the product of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `*` in identifiers is `mul`.
abs : {α : Type u_1} → [Lattice α] → [AddGroup α] → α → α`abs a`, denoted `|a|`, is the absolute value of `a`
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 → PropThe 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 = ∞`.
Real.exp : ℝ → ℝThe real exponential function, defined as the real part of the complex exponential
MeasureTheory.Measure.bind : {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} →
{mβ : MeasurableSpace β} → MeasureTheory.Measure α → (α → MeasureTheory.Measure β) → MeasureTheory.Measure βMonadic bind on `Measure`, only works in the category of measurable spaces and measurable functions. When the function `f` is not measurable the result is not well defined.
Code
lemma memLp_exp_mul (h : HasSubexponentialMGF X V b κ ν) {t : ℝ} (p : ℝ≥0)
(ht : b * |(p : ℝ) * t| ≤ 1) :
MemLp (fun ω ↦ exp (t * X ω)) p (κ ∘ₘ ν)Proof
by
refine ⟨continuous_exp.comp_aestronglyMeasurable (h.aestronglyMeasurable.const_mul t), ?_⟩
by_cases hp0 : p = 0
· simp [hp0]
rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (mod_cast hp0) (by simp)]
simp only [ENNReal.coe_toReal]
have h' := (h.integrable_exp_mul (p * t) ht).2
rw [hasFiniteIntegral_def] at h'
convert! h' using 3 with ω
rw [enorm_eq_ofReal (by positivity), enorm_eq_ofReal (by positivity),
ENNReal.ofReal_rpow_of_nonneg (by positivity), ← exp_mul, mul_comm, ← mul_assoc]
positivityMeaning 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, 66 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.