`MeasureTheory.Measure.dirac_compProd`🔗

Lemma

No docstring.

Types

β : Type u_2mβ : MeasurableSpace βA measurable space is a space equipped with a σ-algebra.
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

κ : ProbabilityTheory.Kernel β ΩA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsSFiniteKernel κA kernel is s-finite if it can be written as the sum of countably many finite kernels.
b : β

Then

(dirac b).compProd κ = map (Prod.mk b) (κ b)

MeasurableSpace : Type u_6 → Type u_6

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

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

A kernel is s-finite if it can be written as the sum of countably many finite kernels.

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)`.

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`.

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

The dirac measure.

MeasureTheory.Measure.compProd : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} → MeasureTheory.Measure α → ProbabilityTheory.Kernel α β → MeasureTheory.Measure (α × β)

The composition-product of a measure and a kernel.

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 β

The 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.

Prod.mk : {α : Type u} → {β : Type v} → α → β → α × β

Constructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`.


Conventions for notations in identifiers:

 * The recommended spelling of `(a, b)` in identifiers is `mk`.

Code

lemma _root_.MeasureTheory.Measure.dirac_compProd {κ : Kernel β Ω} [IsSFiniteKernel κ] (b : β) :
    Measure.dirac b ⊗ₘ κ = (κ b).map (Prod.mk b)

Proof

by
  ext s hs
  rw [Measure.compProd_apply hs, lintegral_dirac' _ (Kernel.measurable_kernel_prodMk_left hs),
    Measure.map_apply measurable_prodMk_left hs]

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, 13 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.

MeasureTheory.Measure.dirac_compProd🔗

`MeasureTheory.Measure.dirac_compProd`🔗