ProbabilityTheory.HasSubexponentialMGF.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
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V : ℝ -
b : ℝ -
μ : 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. -
Y : Ω' → Ω -
X : Ω → ℝ
Assuming
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hY : AEMeasurable Y μA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. -
hX : AEMeasurable X (MeasureTheory.Measure.map Y μ)
Then
HasSubexponentialMGF X V b (MeasureTheory.Measure.map Y μ) ↔ HasSubexponentialMGF (X ∘ Y) V b μMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
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`.
AEMeasurable : {α : Type u_1} →
{β : Type u_2} →
[MeasurableSpace β] →
{_m : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) AEMeasurable._auto_1 → PropA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. A similar notion is `MeasureTheory.NullMeasurable`. That notion is equivalent to `AEMeasurable` if the σ-algebra on the codomain is countably generated, but weaker in general.
MeasureTheory.Measure.map : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure βThe pushforward of a measure. If `f` is not an almost everywhere measurable function, we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way we always have `map f 0 = 0`, and the push-forward of a probability measure is always a probability measure.
Iff : Prop → Prop → PropIf 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.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
Function.comp : {α : Sort u} → {β : Sort v} → {δ : Sort w} → (β → δ) → (α → β) → α → δFunction composition, usually written with the infix operator `∘`. A new function is created from two existing functions, where one function's output is used as input to the other. Examples: * `Function.comp List.reverse (List.drop 2) [3, 2, 4, 1] = [1, 4]` * `(List.reverse ∘ List.drop 2) [3, 2, 4, 1] = [1, 4]` Conventions for notations in identifiers: * The recommended spelling of `∘` in identifiers is `comp`.
Code
lemma map_iff {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {μ : Measure Ω'}
{Y : Ω' → Ω} {X : Ω → ℝ} (hY : AEMeasurable Y μ) (hX : AEMeasurable X (μ.map Y)) :
HasSubexponentialMGF X V b (μ.map Y) ↔ HasSubexponentialMGF (X ∘ Y) V b μProof
by
refine ⟨fun h ↦ .of_map hY h, fun h ↦ ⟨fun t ht ↦ ?_, fun t ht ↦ ?_⟩⟩
· rw [integrable_map_measure (by fun_prop) hY]
exact h.integrable_exp_mul t ht
· rw [mgf_map hY (by fun_prop)]
exact h.mgf_le t htMeaning 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.