Theorems · Theorem · measure theory

generateFrom_pi_eq

∀ {ι : Type u_1} {α : ι → Type u_2} [Finite ι] {C : (i : ι) → Set (Set (α i))},
  (∀ (i : ι), IsCountablySpanning (C i)) →
    MeasurableSpace.pi = MeasurableSpace.generateFrom (Set.univ.pi '' Set.univ.pi C)

The product of generated σ-algebras is the one generated by boxes, if both generating sets are countably spanning.

Defined in: Mathlib.MeasureTheory.MeasurableSpace.Pi
Cited by: 1 results in Mathlib
Foundations: Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Finite

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
MeasurableSpacestatement and proof · cited by 13,106
Set.imagestatement and proof · cited by 5,609
Set.preimageproof · cited by 4,946
Set.univstatement and proof · cited by 3,945
MeasurableSetproof · cited by 3,075
Finitestatement and proof · cited by 3,029
Set.iUnionproof · cited by 2,483
Set.extproof · cited by 2,266
le_antisymmproof · cited by 2,068
Function.updateproof · cited by 502
Set.mem_univproof · cited by 416

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generateFrom_eq_piproof · cited by 3