ProbabilityTheory.iIndepFun.indepFun_of_measurable_iSup_comap
From the authors
A coordinate of an independent family is independent of any function that is measurable with respect to the other coordinates.
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ι : Type u_1 -
Ω : Type u_2mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
β : Type u_3mβ : MeasurableSpace β
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𝓧 : ι → Type u_4(i : ι) → MeasurableSpace (𝓧 i) -
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 : (i : ι) → Ω → 𝓧 i -
S : Set ιA set is a collection of elements of some typeα. -
i : ι -
Y : Ω → β
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hX : iIndepFun X μA family of functions defined on the same spaceΩand taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures… -
hXm : ∀ (i : ι), Measurable (X i)A functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable. -
hi : i ∉ S -
hY : Measurable Y
IndepFun (X i) Y μMeasurableSpace : 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`.
Set : Type u → Type uA set is a collection of elements of some type `α`.
Although `Set` is defined as `α → Prop`, this is an implementation detail which should not be
relied on. Instead, `Set.ofPred` (also written `{x | p x}`) and membership of a set (`∈`) should be
used to convert between sets and predicates.ProbabilityTheory.iIndepFun : {Ω : Type u_1} →
{ι : Type u_2} →
{_mΩ : MeasurableSpace Ω} →
{β : ι → Type u_6} →
[m : (x : ι) → MeasurableSpace (β x)] →
((x : ι) → Ω → β x) → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.iIndepFun._auto_1 → PropA family of functions defined on the same space `Ω` and taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures they generate on `Ω` is independent. For a function `g` with codomain having measurable space structure `m`, the generated measurable space structure is `MeasurableSpace.comap g m`.
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.
Not : Prop → Prop`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`, so if your goal is `¬p` you can use `intro h` to turn the goal into `h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False` and `(hn h).elim` will prove anything. For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic) Conventions for notations in identifiers: * The recommended spelling of `¬` in identifiers is `not`.
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`).
Code
lemma iIndepFun.indepFun_of_measurable_iSup_comap {ι Ω β : Type*} {𝓧 : ι → Type*}
[∀ i, MeasurableSpace (𝓧 i)] {mΩ : MeasurableSpace Ω} {mβ : MeasurableSpace β}
{μ : Measure Ω} {X : ∀ i, Ω → 𝓧 i} (hX : iIndepFun X μ) (hXm : ∀ i, Measurable (X i))
{S : Set ι} {i : ι} (hi : i ∉ S) {Y : Ω → β}
(hY : Measurable[⨆ j ∈ S, MeasurableSpace.comap (X j) inferInstance] Y) :
IndepFun (X i) Y μProof
by
rw [IndepFun_iff_Indep]
refine indep_of_indep_of_le_right ?_ hY.comap_le
have h := indep_iSup_of_disjoint (fun j ↦ (hXm j).comap_le) hX.iIndep (S := {i}) (T := S)
(Set.disjoint_singleton_left.2 hi)
simpa using hMeaning 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, 17 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.