`ProbabilityTheory.hasLaw_of_forall_eventually_eq`🔗

Lemma

From the authors

Let Y n : Ω → Π i, 𝓧 i be random variables with law μ, indexed by a countably generated filter L. If for every ω and i, Y n ω i is eventually equal to Y' ω i along L, then Y' also has law μ.

Types

Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
ι : Type u_3
κ : Type u_5

Given

P : 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.IsFiniteMeasure PA measure μ is called finite if μ univ < ∞.
𝓧 : ι → Type u_4(i : ι) → MeasurableSpace (𝓧 i)
L : Filter κA filter F on a type α is a collection of sets of α which contains the whole α, is upwards-closed, and is stable under intersection.L.NeBotA filter is NeBot if it is not equal to ⊥, or equivalently the empty set does not belong to the filter.L.IsCountablyGeneratedIsCountablyGenerated f means f = generate s for some countable s.
μ : MeasureTheory.Measure ((i : ι) → 𝓧 i)
Y : κ → Ω → (i : ι) → 𝓧 i
Y' : Ω → (i : ι) → 𝓧 i

Assuming

hY : ∀ (n : κ), Measurable (Y n)A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.
hY' : AEMeasurable Y' PA function is almost everywhere measurable if it coincides almost everywhere with a measurable function.
h_law : ∀ (n : κ), HasLaw (Y n) μ PThe predicate HasLaw X μ P registers the fact that the random variable X has law μ under the measure P, in other words that P.map X = μ.
h_lim : ∀ (ω : Ω) (i : ι), ∀ᶠ (n : κ) in L, Y n ω i = Y' ω if.Eventually p or ∀ᶠ x in f, p x mean that {x | p x} ∈ f.

Then

HasLaw Y' μ P

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called finite if `μ univ < ∞`.

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`.

Filter.NeBot : {α : Type u_1} → Filter α → Prop

A filter is `NeBot` if it is not equal to `⊥`, or equivalently the empty set does not belong to
the filter. Bourbaki include this assumption in the definition of a filter but we prefer to have a
`CompleteLattice` structure on `Filter _`, so we use a typeclass argument in lemmas instead.

Filter.IsCountablyGenerated : {α : Type u_1} → Filter α → Prop

`IsCountablyGenerated f` means `f = generate s` for some countable `s`.

Filter : Type u_1 → Type u_1

A filter `F` on a type `α` is a collection of sets of `α` which contains the whole `α`,
is upwards-closed, and is stable under intersection. We do not forbid this collection to be
all sets of `α`.

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.

AEMeasurable : {α : Type u_1} →
  {β : Type u_2} →
    [MeasurableSpace β] →
      {_m : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) AEMeasurable._auto_1 → Prop

A 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.

ProbabilityTheory.HasLaw : {Ω : Type u_1} →
  {𝓧 : Type u_2} →
    {mΩ : MeasurableSpace Ω} →
      {m𝓧 : MeasurableSpace 𝓧} →
        (Ω → 𝓧) → MeasureTheory.Measure 𝓧 → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasLaw._auto_1 → Prop

The predicate `HasLaw X μ P` registers the fact that the random variable `X` has law `μ` under
the measure `P`, in other words that `P.map X = μ`. We also require `X` to be `AEMeasurable`,
to allow for nice interactions with operations on the codomain of `X`. See for instance
`HasLaw.comp`, `IndepFun.hasLaw_mul` and `IndepFun.hasLaw_add`.

Filter.Eventually : {α : Type u_1} → (α → Prop) → Filter α → Prop

`f.Eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in atTop, p x`
means that `p` holds true for sufficiently large `x`.

Eq : {α : Sort u_1} → α → α → Prop

The 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`.

Code

lemma hasLaw_of_forall_eventually_eq [IsFiniteMeasure P] {κ : Type*} {L : Filter κ} [L.NeBot]
    [L.IsCountablyGenerated] {μ : Measure (Π i, 𝓧 i)} {Y : κ → Ω → Π i, 𝓧 i} {Y' : Ω → Π i, 𝓧 i}
    (hY : ∀ n, Measurable (Y n)) (hY' : AEMeasurable Y' P)
    (h_law : ∀ n, HasLaw (Y n) μ P) (h_lim : ∀ ω i, ∀ᶠ n in L, Y n ω i = Y' ω i) :
    HasLaw Y' μ P where
  aemeasurable

Proof

hY'
  map_eq := by
    refine ext_of_generate_finite (measurableCylinders _) generateFrom_measurableCylinders.symm
      isPiSystem_measurableCylinders (fun s hs ↦ ?_) ?_
    · obtain ⟨I, S, hS, rfl⟩ := (mem_measurableCylinders s).1 hs
      rw [Measure.map_apply_of_aemeasurable hY' (hS.cylinder _)]
      have h_tendsto : Tendsto (fun n ↦ P (Y n ⁻¹' cylinder I S)) L
          (𝓝 (P (Y' ⁻¹' cylinder I S))) := by
        refine tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure L P
          (fun n ↦ (hS.cylinder _).preimage (hY n)) fun ω ↦ ?_
        have h_ev : ∀ᶠ n in L, ∀ i ∈ I, Y n ω i = Y' ω i :=
          (eventually_all_finset I).2 fun i _ ↦ h_lim ω i
        filter_upwards [h_ev] with n hn
        simp only [Set.mem_preimage, mem_cylinder]
        have : I.restrict (Y n ω) = I.restrict (Y' ω) := funext fun i ↦ hn i i.2
        rw [this]
      have h_const : Tendsto (fun n ↦ P (Y n ⁻¹' cylinder I S)) L (𝓝 (μ (cylinder I S))) := by
        have : (fun n ↦ P (Y n ⁻¹' cylinder I S)) = fun _ ↦ μ (cylinder I S) := by
          funext n
          rw [← Measure.map_apply (hY n) (hS.cylinder _), (h_law n).map_eq]
        rw [this]
        exact tendsto_const_nhds
      exact tendsto_nhds_unique h_tendsto h_const
    · obtain ⟨n⟩ := L.nonempty_of_neBot
      rw [Measure.map_apply_of_aemeasurable hY' MeasurableSet.univ, Set.preimage_univ,
        ← (h_law n).map_eq,
        Measure.map_apply (hY n) MeasurableSet.univ, Set.preimage_univ]

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

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 12 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.hasLaw_of_forall_eventually_eq🔗

`ProbabilityTheory.hasLaw_of_forall_eventually_eq`🔗