MeasurableSpace.iSup_comap_restrictFin
From the authors
The σ-algebras of the projections of ℕ → E on the first n coordinates generate the
product σ-algebra.
-
E : Type u_1mE : MeasurableSpace EA measurable space is a space equipped with a σ-algebra.
⨆ n, MeasurableSpace.comap (fun f i => f ↑i) pi = piMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
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`.MeasurableSpace.comap : {α : Type u_1} → {β : Type u_2} → (α → β) → MeasurableSpace β → MeasurableSpace αThe reverse image of a measurable space under a function. `comap f m` contains the sets `s : Set α` such that `s` is the `f`-preimage of a measurable set in `β`.
MeasurableSpace.pi : {δ : Type u_4} → {X : δ → Type u_6} → [m : (a : δ) → MeasurableSpace (X a)] → MeasurableSpace ((a : δ) → X a)Code
lemma iSup_comap_restrictFin {E : Type*} [mE : MeasurableSpace E] :
⨆ n : ℕ, MeasurableSpace.comap (fun f : ℕ → E ↦ fun i : Fin n ↦ f i)
MeasurableSpace.pi = MeasurableSpace.piProof
by
refine le_antisymm
(iSup_le fun n ↦ (Measurable.of_eval fun _ ↦ measurable_pi_apply _).comap_le)
(iSup_le fun i ↦ le_iSup_of_le (i + 1) ?_)
have : (fun f : ℕ → E ↦ f i) =
(fun h : Fin (i + 1) → E ↦ h ⟨i, i.lt_succ_self⟩) ∘
fun f : ℕ → E ↦ fun j : Fin (i + 1) ↦ f j := rfl
rw [this, ← MeasurableSpace.comap_comp]
exact MeasurableSpace.comap_mono (measurable_pi_apply _).comap_le
New in v4.34.0-rc2-49-gc33199e (2026-09-09), and its
meaning has not changed since. Its file was edited 2026-09-10 (357e9dd, “Update Mathlib”) without changing what it means.
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.