ProbabilityTheory.HasSubexponentialMGF.congr_identDistrib
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
-
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 -
μ' : MeasureTheory.Measure Ω' -
Y : Ω' → ℝ
Assuming
-
hX : HasSubexponentialMGF X V b μXhas a sub-exponential moment generating function with parameters(V, b): for everytwithb * |t| ≤ 1,exp (t * X)is integrable andmgf X μ t ≤ exp (V * t ^ 2 / 2). -
hXY : IdentDistrib X Y μ μ'Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide.
Then
HasSubexponentialMGF Y V b μ'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`.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
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
ProbabilityTheory.IdentDistrib : {α : Type u_1} →
{β : Type u_2} →
{γ : Type u_3} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] →
[MeasurableSpace γ] →
(α → γ) →
(β → γ) →
autoParam (MeasureTheory.Measure α) ProbabilityTheory.IdentDistrib._auto_1 →
autoParam (MeasureTheory.Measure β) ProbabilityTheory.IdentDistrib._auto_3 → Pro…Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition.
Code
lemma congr_identDistrib {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {μ' : Measure Ω'}
{Y : Ω' → ℝ} (hX : HasSubexponentialMGF X V b μ) (hXY : IdentDistrib X Y μ μ') :
HasSubexponentialMGF Y V b μ'Proof
by rw [← id_map_iff hXY.aemeasurable_fst] at hX rwa [← id_map_iff hXY.aemeasurable_snd, ← hXY.map_eq]
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, 54 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.