`ProbabilityTheory.Kernel.hasSubexponentialMGF_congr`🔗

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
Y : Ω → ℝ

Assuming

h : X =ᵐ[ν.bind ⇑κ] YTwo functions f and g are *eventually equal* along a filter l if the set of x such that f x = g x belongs to l.

Then

HasSubexponentialMGF X V b κ ν ↔ HasSubexponentialMGF Y 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.

Filter.EventuallyEq : {α : Type u_1} → {β : Type u_2} → Filter α → (α → β) → (α → β) → Prop

Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
`f x = g x` belongs to `l`.

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 β

Monadic 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.

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

Code

lemma _root_.ProbabilityTheory.Kernel.hasSubexponentialMGF_congr {Y : Ω → ℝ}
    (h : X =ᵐ[κ ∘ₘ ν] Y) :
    HasSubexponentialMGF X V b κ ν ↔ HasSubexponentialMGF Y V b κ ν

Proof

⟨fun hX ↦ congr hX h, fun hY ↦ congr hY (ae_eq_symm h)⟩

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, 62 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_congr🔗

`ProbabilityTheory.Kernel.hasSubexponentialMGF_congr`🔗