`Learning.Kernel.eq_const_dirac_unit`🔗

Lemma

From the authors

Every Markov kernel with codomain Unit is the constant kernel at Measure.dirac ().

Types

α : Type u_5mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra.

Given

κ : ProbabilityTheory.Kernel α UnitA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel κA kernel is a Markov kernel if every measure in its image is a probability measure.

Then

κ = ProbabilityTheory.Kernel.const α (MeasureTheory.Measure.dirac ())

MeasurableSpace : Type u_6 → Type u_6

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

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A 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)`.

Unit : Type

The canonical type with one element. This element is written `()`.

`Unit` has a number of uses:
* It can be used to model control flow that returns from a function call without providing other
  information.
* Monadic actions that return `Unit` have side effects without computing values.
* In polymorphic types, it can be used to indicate that no data is to be stored in a particular
  field.

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`.

ProbabilityTheory.Kernel.const : (α : Type u_4) →
  {β : Type u_5} →
    [inst : MeasurableSpace α] → {x : MeasurableSpace β} → MeasureTheory.Measure β → ProbabilityTheory.Kernel α β

Constant kernel, which always returns the same measure.

MeasureTheory.Measure.dirac : {α : Type u_1} → [inst : MeasurableSpace α] → α → MeasureTheory.Measure α

The dirac measure.

Unit.unit : Unit

The only element of the unit type.

It can be written as an empty tuple: `()`.

Code

lemma Kernel.eq_const_dirac_unit {α : Type*} {mα : MeasurableSpace α} (κ : Kernel α Unit)
    [IsMarkovKernel κ] :
    κ = Kernel.const α (Measure.dirac ())

Proof

by
  ext a s hs
  rw [Kernel.const_apply]
  rcases Set.eq_empty_or_nonempty s with rfl | ⟨u, hu⟩
  · simp
  · have hs_univ : s = Set.univ := Set.eq_univ_of_forall fun x ↦ by rwa [Subsingleton.elim x u]
    simp [hs_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, 9 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.

Learning.Kernel.eq_const_dirac_unit🔗

`Learning.Kernel.eq_const_dirac_unit`🔗