`InformationTheory.klDiv_map_measurableEmbedding`🔗

Lemma

From the authors

The Kullback–Leibler divergence is invariant under measurable embeddings.

Types

α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra.
β : Type u_2mβ : MeasurableSpace β

Given

μ : MeasureTheory.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.MeasureTheory.IsFiniteMeasure μA measure μ is called finite if μ univ < ∞.
ν : MeasureTheory.Measure αMeasureTheory.IsFiniteMeasure ν
f : α → β

Assuming

hf : MeasurableEmbedding fA map f : α → β is called a *measurable embedding* if it is injective, measurable, and sends measurable sets to measurable sets.

Then

klDiv (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map f ν) = klDiv μ ν

MeasurableSpace : Type u_6 → Type u_6

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

MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called finite if `μ univ < ∞`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A 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`.

MeasurableEmbedding : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A map `f : α → β` is called a *measurable embedding* if it is injective, measurable, and sends
measurable sets to measurable sets. The latter assumption can be replaced with “`f` has measurable
inverse `g : Set.range f → α`”, see `MeasurableEmbedding.measurable_rangeSplitting`,
`MeasurableEmbedding.of_measurable_inverse_range`, and
`MeasurableEmbedding.of_measurable_inverse`.

One more interpretation: `f` is a measurable embedding if it defines a measurable equivalence to its
range and the range is a measurable set. One implication is formalized as
`MeasurableEmbedding.equivRange`; the other one follows from
`MeasurableEquiv.measurableEmbedding`, `MeasurableEmbedding.subtype_coe`, and
`MeasurableEmbedding.comp`.

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`.

InformationTheory.klDiv : {α : Type u_2} → {mα : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → ENNReal

Kullback-Leibler divergence between two measures.

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.

Code

lemma klDiv_map_measurableEmbedding (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    {f : α → β} (hf : MeasurableEmbedding f) :
    klDiv (μ.map f) (ν.map f) = klDiv μ ν

Proof

by
  refine le_antisymm (klDiv_map_le μ ν hf.measurable) ?_
  rcases isEmpty_or_nonempty α with hα | hα
  · simp [μ.eq_zero_of_isEmpty, ν.eq_zero_of_isEmpty]
  have h := klDiv_map_le (μ.map f) (ν.map f) hf.measurable_invFun
  rwa [Measure.map_map hf.measurable_invFun hf.measurable,
    Measure.map_map hf.measurable_invFun hf.measurable, hf.leftInverse_invFun.comp_eq_id,
    Measure.map_id, Measure.map_id] at h

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, 8 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.

InformationTheory.klDiv_map_measurableEmbedding🔗

`InformationTheory.klDiv_map_measurableEmbedding`🔗