`ProbabilityTheory.Kernel.HasSubexponentialMGF.id_map_iff`🔗

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 space α to another measurable space β is a measurable function κ : α → Measure β.
X : Ω → ℝ
V : ℝ
b : ℝ

Assuming

hX : Measurable XA function f between measurable spaces is measurable if the preimage of every measurable set is measurable.

Then

HasSubexponentialMGF id V b (κ.map X) ν ↔ HasSubexponentialMGF X V b κ ν

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`.

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 : Type

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

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

Iff : Prop → Prop → Prop

If and only if, or logical bi-implication. `a ↔ b` means that `a` implies `b` and vice versa.
By `propext`, this implies that `a` and `b` are equal and hence any expression involving `a`
is equivalent to the corresponding expression with `b` instead.


Conventions for notations in identifiers:

 * The recommended spelling of `↔` in identifiers is `iff`.

 * The recommended spelling of `<->` in identifiers is `iff` (prefer `↔` over `<->`).

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 → Prop

A 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

id : {α : Sort u} → α → α

The identity function. `id` takes an implicit argument `α : Sort u`
(a type in any universe), and an argument `a : α`, and returns `a`.

Although this may look like a useless function, one application of the identity
function is to explicitly put a type on an expression. If `e` has type `T`,
and `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean
knows that this expression has type `T'` rather than `T`. This can make a
difference for typeclass inference, since `T` and `T'` may have different
typeclass instances on them. `show T' from e` is sugar for an `@id T' e`
expression.

ProbabilityTheory.Kernel.map : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} →
        {γ : Type u_4} →
          [inst : MeasurableSpace γ] → ProbabilityTheory.Kernel α β → (β → γ) → ProbabilityTheory.Kernel α γ

The pushforward of a kernel along a function.
If the function is not measurable, we use zero instead. This choice of junk
value ensures that typeclass inference can infer that the `map` of a kernel
satisfying `IsZeroOrMarkovKernel` again satisfies this property.

Code

lemma id_map_iff (hX : Measurable X) :
    HasSubexponentialMGF id V b (κ.map X) ν ↔ HasSubexponentialMGF X V b κ ν

Proof

by
  refine ⟨fun h ↦ ?_, fun h ↦ ⟨fun t ht ↦ ?_, ?_⟩⟩
  · change HasSubexponentialMGF (id ∘ X) V b κ ν
    exact .of_map hX h
  · rw [← Kernel.deterministic_comp_eq_map hX, ← Measure.comp_assoc,
      Measure.deterministic_comp_eq_map, integrable_map_measure (by fun_prop) hX.aemeasurable]
    exact h.integrable_exp_mul t ht
  · simpa [Kernel.map_apply _ hX, mgf_id_map hX.aemeasurable] using h.mgf_le

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.

ProbabilityTheory.Kernel.HasSubexponentialMGF.id_map_iff🔗

`ProbabilityTheory.Kernel.HasSubexponentialMGF.id_map_iff`🔗