Theorems · Theorem · measure theory

generateFrom_eq_pi

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

If C and D generate the σ-algebras on α resp. β, then rectangles formed by C and D generate the σ-algebra on α × β.

Defined in: Mathlib.MeasureTheory.MeasurableSpace.Pi
Cited by: 3 results in Mathlib
Foundations: Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Finite MeasurableSpace

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Cites9

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.univstatement and proof · cited by 3,945
Finitestatement and proof · cited by 3,029
Set.pistatement and proof · cited by 405
MeasurableSpace.generateFromstatement and proof · cited by 172
IsCountablySpanningstatement and proof · cited by 12
generateFrom_pi_eqproof · cited by 1

Cited by3

Results whose statement or proof uses this declaration.

MeasureTheory.Measure.pi_eq_generateFromproof · cited by 2
generateFrom_piproof · cited by 2
Measure.ext_of_lintegral_prod_mul_prod_boundedContinuousFunctionproof · cited by 1