ProbabilityTheory.HasSubexponentialMGF.integral_eq_zero
Lemma
From the authors
A sub-exponential random variable is centered.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
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.MeasureTheory.IsProbabilityMeasure μA measureμis called a probability measure ifμ univ = 1. -
X : Ω → ℝ -
V : ℝ -
b : ℝ
Assuming
Then
∫ (x : Ω), X x ∂μ = 0MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called a probability measure if `μ univ = 1`.
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
Eq : {α : Sort u_1} → α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.MeasureTheory.integral : {α : Type u_6} →
{G : Type u_7} →
[inst : NormedAddCommGroup G] → [NormedSpace ℝ G] → {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → G) → GThe Bochner integral
Code
lemma integral_eq_zero (h : HasSubexponentialMGF X V b μ) [IsProbabilityMeasure μ] : μ[X] = 0
Proof
by
have hd : HasDerivAt (fun t ↦ mgf X μ t - exp (V * t ^ 2 / 2)) (μ[X] - 0) 0 := by
have h1 : HasDerivAt (mgf X μ) μ[X] 0 := by
simpa using hasDerivAt_mgf h.zero_mem_interior_integrableExpSet
have h2 : HasDerivAt (fun t : ℝ ↦ exp (V * t ^ 2 / 2)) 0 0 := by
have := (((hasDerivAt_pow 2 (0 : ℝ)).const_mul V).div_const 2).exp
simpa using this
exact h1.sub h2
have hmax : IsLocalMax (fun t ↦ mgf X μ t - exp (V * t ^ 2 / 2)) 0 := by
filter_upwards [((by fun_prop : Continuous fun t : ℝ ↦ b * |t|).tendsto 0).eventually_le_const
(by simp : b * |(0 : ℝ)| < 1)] with t ht
simp only [mgf_zero', probReal_univ, zero_pow two_ne_zero, mul_zero, zero_div, exp_zero,
sub_self]
linarith [h.mgf_le t ht]
simpa using hmax.hasDerivAt_eq_zero hdMeaning 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.