ProbabilityTheory.Kernel.sigma_apply
Lemma
No docstring.
Types
-
α : Type u_1 -
γ : Type u_2mγ : MeasurableSpace γA measurable space is a space equipped with a σ-algebra.
Given
-
β : α → Type u_3(a : α) → MeasurableSpace (β a) -
κ : (a : α) → Kernel (β a) γA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β. -
x : (a : α) × β a
Then
(sigma κ) x = (κ x.fst) x.sndMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
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)`.
Sigma : {α : Type u} → (α → Type v) → Type (max u v)Dependent pairs, in which the second element's type depends on the value of the first element. The type `Sigma β` is typically written `Σ a : α, β a` or `(a : α) × β a`. Although its values are pairs, `Sigma` is sometimes known as the *dependent sum type*, since it is the type level version of an indexed summation.
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`.ProbabilityTheory.Kernel.sigma : {α : Type u_1} →
{γ : Type u_2} →
{β : α → Type u_3} →
[inst : (a : α) → MeasurableSpace (β a)] →
{mγ : MeasurableSpace γ} →
((a : α) → ProbabilityTheory.Kernel (β a) γ) → ProbabilityTheory.Kernel ((a : α) × β a) γThe kernel on `Σ a, β a` which is `κ a` on the fiber `β a`.Go to its page
Sigma.fst : {α : Type u} → {β : α → Type v} → Sigma β → αThe first component of a dependent pair.
Sigma.snd : {α : Type u} → {β : α → Type v} → (self : Sigma β) → β self.fstThe second component of a dependent pair. Its type depends on the first component.
Code
lemma sigma_apply (κ : (a : α) → Kernel (β a) γ) (x : Σ a, β a) : sigma κ x = κ x.1 x.2
Proof
rfl
New in v4.34.0-rc2-82-ga6c27a7 (2026-09-13), and its meaning has not changed since.
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, 11 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.