ProbabilityTheory.cond_prod_univ
Lemma
From the authors
Conditioning a product measure on an event of the first coordinate amounts to conditioning the first measure.
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.SFinite μA measure is called s-finite if it is a countable sum of finite measures. -
ν : MeasureTheory.Measure βMeasureTheory.IsProbabilityMeasure νA measureμis called a probability measure ifμ univ = 1. -
s : Set αA set is a collection of elements of some typeα.
Then
(μ.prod ν)[|s ×ˢ Set.univ] = μ[|s].prod νMeasurableSpace : 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`.
MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called a probability measure if `μ univ = 1`.
Set : Type u → Type uA set is a collection of elements of some type `α`.
Although `Set` is defined as `α → Prop`, this is an implementation detail which should not be
relied on. Instead, `Set.ofPred` (also written `{x | p x}`) and membership of a set (`∈`) should be
used to convert between sets and predicates.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.prod : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → MeasureTheory.Measure α → MeasureTheory.Measure β → MeasureTheory.Measure (α × β)The binary product of measures. They are defined for arbitrary measures, but we basically prove all properties under the assumption that at least one of them is s-finite.
ProbabilityTheory.cond : {Ω : Type u_1} → {m : MeasurableSpace Ω} → MeasureTheory.Measure Ω → Set Ω → MeasureTheory.Measure ΩThe conditional probability measure of measure `μ` on set `s` is `μ` restricted to `s` and scaled by the inverse of `μ s` (to make it a probability measure): `(μ s)⁻¹ • μ.restrict s`.
SProd.sprod : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : SProd α β γ] → α → β → γThe Cartesian product `s ×ˢ t` is the set of `(a, b)` such that `a ∈ s` and `b ∈ t`.
Set.univ : {α : Type u} → Set αThe universal set on a type `α` is the set containing all elements of `α`. This is conceptually the "same as" `α` (in set theory, it is actually the same), but type theory makes the distinction that `α` is a type while `Set.univ` is a term of type `Set α`. `Set.univ` can itself be coerced to a type `↥Set.univ` which is in bijection with (but distinct from) `α`.
Code
lemma cond_prod_univ {μ : Measure α} [SFinite μ] {ν : Measure β} [IsProbabilityMeasure ν]
(s : Set α) :
(μ.prod ν)[|s ×ˢ Set.univ] = (μ[|s]).prod νProof
by
simp only [cond, Measure.prod_prod, measure_univ, mul_one, Measure.prod_smul_left,
← Measure.prod_restrict, Measure.restrict_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, 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.