ProbabilityTheory.Kernel.eq_trajMeasureFin_map
Lemma
No docstring.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
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.IsProbabilityMeasure PA measureμis called a probability measure ifμ univ = 1. -
X : ℕ → Type u_2(n : ℕ) → MeasurableSpace (X n) -
κ' : (n : ℕ) → Kernel ((i : Fin n) → X ↑i) (X n)A kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.∀ (n : ℕ), IsMarkovKernel (κ' n)A kernel is a Markov kernel if every measure in its image is a probability measure. -
Y : (n : ℕ) → Ω → X n -
N : ℕ
Assuming
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hY_meas : ∀ (n : ℕ), Measurable (Y n)A functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable. -
h_condDistrib : ∀ n < N, HasCondDistrib (Y n) (fun ω i => Y (↑i) ω) (κ' n) PPredicate stating that the conditional distribution ofYgivenXunder the measurePis equal to the kernelκ.
Then
MeasureTheory.Measure.map (fun ω i => Y (↑i) ω) P = MeasureTheory.Measure.map (fun x i => x ↑i) (trajMeasureFin κ')MeasurableSpace : 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`.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA kernel is a Markov kernel if every measure in its image is a probability measure.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
Fin : ℕ → TypeNatural numbers less than some upper bound. In particular, a `Fin n` is a natural number `i` with the constraint that `i < n`. It is the canonical type with `n` elements.
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.HasCondDistrib : {Ω : Type u_1} →
{𝓧 : Type u_2} →
{𝓨 : Type u_3} →
{mΩ : MeasurableSpace Ω} →
{m𝓧 : MeasurableSpace 𝓧} →
{m𝓨 : MeasurableSpace 𝓨} → (Ω → 𝓨) → (Ω → 𝓧) → ProbabilityTheory.Kernel 𝓧 𝓨 → MeasureTheory.Measure Ω → PropPredicate stating that the conditional distribution of `Y` given `X` under the measure `P` is equal to the kernel `κ`.
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.Measure.map : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure βThe pushforward of a measure. If `f` is not an almost everywhere measurable function, we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way we always have `map f 0 = 0`, and the push-forward of a probability measure is always a probability measure.
ProbabilityTheory.Kernel.trajMeasureFin : {X : ℕ → Type u_2} →
[inst : (n : ℕ) → MeasurableSpace (X n)] →
(κ' : (n : ℕ) → ProbabilityTheory.Kernel ((i : Fin n) → X ↑i) (X n)) →
[∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (κ' n)] → MeasureTheory.Measure ((n : ℕ) → X n)Measure on trajectories `Π n, X n` built from kernels `κ' n : Kernel (Π i : Fin n, X i) (X n)` describing the law of the coordinate `n` given the `n` previous coordinates. The initial measure is `κ' 0 default`.Go to its page
Code
lemma eq_trajMeasureFin_map [IsProbabilityMeasure P]
{Y : (n : ℕ) → Ω → X n} (hY_meas : ∀ n, Measurable (Y n)) {N : ℕ}
(h_condDistrib : ∀ n < N, HasCondDistrib (Y n) (fun ω (i : Fin n) ↦ Y i ω) (κ' n) P) :
P.map (fun ω (i : Fin N) ↦ Y i ω) = (trajMeasureFin κ').map (fun x (i : Fin N) ↦ x i)Proof
by
cases N with
| zero =>
rw [show (fun ω (i : Fin 0) ↦ Y i ω) = fun _ ↦ default from
funext fun _ ↦ Unique.eq_default _,
show (fun (x : Π n, X n) (i : Fin 0) ↦ x i) = fun _ ↦ default from
funext fun _ ↦ Unique.eq_default _,
Measure.map_const, Measure.map_const, measure_univ, measure_univ]
| succ N =>
have h0 : HasLaw (Y 0) (κ' 0 default) P := by
have h := h_condDistrib 0 (by omega)
rw [show (fun ω (i : Fin 0) ↦ Y i ω) = fun _ ↦ default from
funext fun _ ↦ Unique.eq_default _] at h
exact h.hasLaw_of_const'
have h := eq_trajMeasure_map_frestrictLe (κ := iicOfFin κ') h0 (N := N) fun n hn ↦
(h_condDistrib (n + 1) (by omega)).measurableEquiv_comp_right
(MeasurableEquiv.finSuccPiIic X n)
have h1 : (fun ω (i : Fin (N + 1)) ↦ Y i ω) =
(MeasurableEquiv.finSuccPiIic X N).symm ∘ (fun ω (n : Iic N) ↦ Y n ω) := rfl
rw [h1, ← Measure.map_map (MeasurableEquiv.measurable _) (by fun_prop), h, trajMeasureFin_def,
Measure.map_map (MeasurableEquiv.measurable _) (by fun_prop),
MeasurableEquiv.finSuccPiIic_symm_comp_frestrictLe]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: 3 project declarations, 46 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.