ProbabilityTheory.indepFun_cond_preimage_singleton_left
From the authors
Under μ conditioned on the event X = b, the random variable X is almost surely constant,
hence independent of any other random variable.
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α : Type u_6mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_7mβ : MeasurableSpace βMeasurableSingletonClass βA typeclass mixin forMeasurableSpaces such that each singleton is measurable. -
γ : Type u_8mγ : MeasurableSpace γ
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μ : MeasureTheory.Measure αA 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 : α → β -
b : β -
Y : α → γ
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hX : Measurable XA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable.
IndepFun X Y μ[|X ⁻¹' {b}]MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasurableSingletonClass : (α : Type u_6) → [MeasurableSpace α] → PropA typeclass mixin for `MeasurableSpace`s such that each singleton is measurable.
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`.
Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
ProbabilityTheory.IndepFun : {Ω : Type u_1} →
{β : Type u_6} →
{γ : Type u_7} →
{_mΩ : MeasurableSpace Ω} →
[MeasurableSpace β] →
[MeasurableSpace γ] →
(Ω → β) → (Ω → γ) → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.IndepFun._auto_1 → PropTwo functions are independent if the two measurable space structures they generate are independent. For a function `f` with codomain having measurable space structure `m`, the generated measurable space structure is `MeasurableSpace.comap f m`. We use the notation `f ⟂ᵢ[μ] g` for `IndepFun f g μ` (scoped in `ProbabilityTheory`).
ProbabilityTheory.cond : {Ω : Type u_1} → {m : MeasurableSpace Ω} → MeasureTheory.Measure Ω → Set Ω → MeasureTheory.Measure ΩThe conditional probability measure of measure `μ` on set `s` is `μ` restricted to `s` and scaled by the inverse of `μ s` (to make it a probability measure): `(μ s)⁻¹ • μ.restrict s`.
Set.preimage : {α : Type u} → {β : Type v} → (α → β) → Set β → Set αThe preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`, is the set of `x : α` such that `f x ∈ s`.
Singleton.singleton : {α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β`singleton x` is a collection with the single element `x` (notation: `{x}`).
Conventions for notations in identifiers:
* The recommended spelling of `{x}` in identifiers is `singleton`.Code
lemma indepFun_cond_preimage_singleton_left {α β γ : Type*} {mα : MeasurableSpace α}
{mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSingletonClass β] {μ : Measure α}
{X : α → β} (hX : Measurable X) (b : β) (Y : α → γ) :
X ⟂ᵢ[μ[|X ⁻¹' {b}]] YProof
(indepFun_const_left b Y).congr
(ae_cond_of_forall_mem (hX (measurableSet_singleton b)) fun x hx ↦ (hx : X x = b).symm)
Filter.EventuallyEq.rflMeaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), the 2th recorded change.
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, 10 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.