Theorems · Theorem · measure theory

generateFrom_prod_eq

∀ {α : Type u_3} {β : Type u_4} {C : Set (Set α)} {D : Set (Set β)},
  IsCountablySpanning C →
    IsCountablySpanning D →
      Prod.instMeasurableSpace = MeasurableSpace.generateFrom (Set.image2 (fun x1 x2 => x1 ×ˢ x2) C D)

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

Defined in: Mathlib.MeasureTheory.MeasurableSpace.Prod
Cited by: 1 results in Mathlib
Foundations: Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites26

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.imageproof · cited by 5,609
Set.preimageproof · cited by 4,946
Set.univproof · cited by 3,945
MeasurableSetproof · cited by 3,075
Set.iUnionproof · cited by 2,483
le_antisymmproof · cited by 2,068
SProd.sprodstatement and proof · cited by 1,750
Set.image2statement and proof · cited by 311
MeasurableSpace.generateFromstatement and proof · cited by 172
MeasurableSet.interproof · cited by 167

Cited by1

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generateFrom_eq_prodproof · cited by 5