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[Merged by Bors] - feat(measure_theory/covering/vitali): Vitali covering theorems #9680
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| by_cases t_eq_empty : t = {∅}, | ||
| { rw t_eq_empty, | ||
| refine ⟨{∅}, subset.refl _, _⟩, |
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| by_cases t_eq_empty : t = {∅}, | |
| { rw t_eq_empty, | |
| refine ⟨{∅}, subset.refl _, _⟩, | |
| rcases eq_or_ne t {∅} with (rfl|t_ne_empty), | |
| { refine ⟨{∅}, subset.refl _, _⟩, |
Also, you can rule out t.subsingleton instead of t = ∅ in the first rcases.
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I tried, but it did not turn out really shorter or clearer, so in the end I came back to my original version. If you see a way which is shorter and clearer, don't hesitate to push directly to the branch!
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The topological and measurable Vitali covering theorems.
* topological version: if a space is covered by balls `(B (x_i, r_i))_{i \in I}`, one can extract a disjointed subfamily indexed by `J` such that the space is covered by the balls `B (x_j, 5 r_j)`.
* measurable version: if additionally the measure has a doubling-like property, and the covering contains balls of arbitrarily small radius at every point, then the disjointed subfamily one obtains above covers almost all the space.
These two statements are particular cases of more general statements that are proved in this PR.
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The topological and measurable Vitali covering theorems.
(B (x_i, r_i))_{i \in I}, one can extract a disjointed subfamily indexed byJsuch that the space is covered by the ballsB (x_j, 5 r_j).These two statements are particular cases of more general statements that are proved in this PR.