MeasureTheory.Measure.map_compProd_comap
Lemma
From the authors
Transporting μ ⊗ₘ η.comap f along f in the first coordinate gives μ.map f ⊗ₘ η.
Types
-
α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2mβ : MeasurableSpace β -
γ : Type u_3mγ : MeasurableSpace γ
Given
-
μ : Measure αA 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.SFinite μA measure is called s-finite if it is a countable sum of finite measures. -
η : ProbabilityTheory.Kernel β γA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.ProbabilityTheory.IsSFiniteKernel ηA kernel is s-finite if it can be written as the sum of countably many finite kernels. -
f : α → β
Assuming
-
hf : Measurable fA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable.
Then
map (fun p => (f p.1, p.2)) (μ.compProd (η.comap f hf)) = (map f μ).compProd ηMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.SFinite : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure is called s-finite if it is a countable sum of finite measures.
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`.
ProbabilityTheory.IsSFiniteKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA 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)`.
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.
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.
Prod.mk : {α : Type u} → {β : Type v} → α → β → α × βConstructs 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`.
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.
ProbabilityTheory.Kernel.comap : {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} →
{mβ : MeasurableSpace β} →
{γ : Type u_4} →
{mγ : MeasurableSpace γ} →
ProbabilityTheory.Kernel α β → (g : γ → α) → Measurable g → ProbabilityTheory.Kernel γ βPullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`. We include measurability in the assumptions instead of using junk values to make sure that typeclass inference can infer that the `comap` of a Markov kernel is again a Markov kernel.
Code
lemma _root_.MeasureTheory.Measure.map_compProd_comap (μ : Measure α) [SFinite μ]
(η : Kernel β γ) [IsSFiniteKernel η] {f : α → β} (hf : Measurable f) :
(μ ⊗ₘ η.comap f hf).map (fun p : α × γ ↦ (f p.1, p.2)) = μ.map f ⊗ₘ ηProof
by
ext s hs
rw [Measure.map_apply (by fun_prop) hs, Measure.compProd_apply (hs.preimage (by fun_prop)),
Measure.compProd_apply hs, lintegral_map (Kernel.measurable_kernel_prodMk_left hs) hf]
rflMeaning 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, 15 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.