[Merged by Bors] - feat(probability_theory/independence): if a family of pi-systems is independent, then so are the generated measurable spaces #9387
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…ity/mathlib into independence_1
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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src/probability/independence.lean
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| @@ -309,6 +308,119 @@ begin | |||
| exact indep_sets.indep_aux h2 hp2 hpm2 hyp ht ht2, | |||
| end | |||
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| variables {α ι : Type*} {m0 : measurable_space α} {μ : measure α} | |||
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| lemma pi_system_indep_insert {π : ι → set (set α)} {a : ι} {S : finset ι} | |||
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The name of this lemma does not really match the symbols in the statement of the lemma.
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…ndependent, then so are the generated measurable spaces (#9387) The main result in this PR is `Indep_sets.Indep`: if π-systems are independent as sets of sets, then the measurable space structures they generate are independent. We already had a version of this for two pi-systems instead of a family. In order to prove this, and as preparation for a next PR about Kolmogorov's 0-1 law, a definition `pi_Union_Inter` is introduced to build a particular pi-system from a family of pi-systems. Co-authored-by: Rémy Degenne <remydegenne@gmail.com> Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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…ndependent, then so are the generated measurable spaces (#9387) The main result in this PR is `Indep_sets.Indep`: if π-systems are independent as sets of sets, then the measurable space structures they generate are independent. We already had a version of this for two pi-systems instead of a family. In order to prove this, and as preparation for a next PR about Kolmogorov's 0-1 law, a definition `pi_Union_Inter` is introduced to build a particular pi-system from a family of pi-systems. Co-authored-by: Rémy Degenne <remydegenne@gmail.com> Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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The main result in this PR is
Indep_sets.Indep: if π-systems are independent as sets of sets, then themeasurable space structures they generate are independent. We already had a version of this for two pi-systems instead of a family.
In order to prove this, and as preparation for a next PR about Kolmogorov's 0-1 law, a definition
pi_Union_Interis introduced to build a particular pi-system from a family of pi-systems.generate_from_monois moved frommeasurable_space.leantomeasurable_space_def.leanbecause I need it in thepi_system.leanfile.