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[Merged by Bors] - feat(measure_theory/order/upper_lower): Order-connected sets in ℝⁿ are measurable
#16976
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YaelDillies
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upper sets form a monoid upper set of open is open antichains have measure zero
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ℝⁿ are measurableℝⁿ are measurable
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…are measurable (#16976) Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. Co-authored-by: @JasonKYi
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* Add `upperBounds_closure`, `lowerBounds_closure`, `bddAbove_closure`, `bddBelow_closure`. * Add `IsAntichain.interior_eq_empty`. * Generalize `nhds_left'_le_nhds_ne` and `nhds_right'_le_nhds_ne` to a `Preorder`. Partly forward-ports leanprover-community/mathlib3#16976
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Feb 18, 2024
* Add `upperBounds_closure`, `lowerBounds_closure`, `bddAbove_closure`, `bddBelow_closure`. * Add `IsAntichain.interior_eq_empty`. * Generalize `nhds_left'_le_nhds_ne` and `nhds_right'_le_nhds_ne` to a `Preorder`. Partly forward-ports leanprover-community/mathlib3#16976
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Mar 22, 2024
* Add `upperBounds_closure`, `lowerBounds_closure`, `bddAbove_closure`, `bddBelow_closure`. * Add `IsAntichain.interior_eq_empty`. * Generalize `nhds_left'_le_nhds_ne` and `nhds_right'_le_nhds_ne` to a `Preorder`. Partly forward-ports leanprover-community/mathlib3#16976
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Partially forward-port leanprover-community/mathlib3#16976
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
utensil
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
Louddy
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
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Partially forward-port leanprover-community/mathlib3#16976 Also fix an unused argument that somehow made it to master, likely from #10661.
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* commit '65a1391a0106c9204fe45bc73a039f056558cb83': (12443 commits)
feat(data/{list,multiset,finset}/*): `attach` and `filter` lemmas (leanprover-community#18087)
feat(combinatorics/simple_graph): More clique lemmas (leanprover-community#19203)
feat(measure_theory/order/upper_lower): Order-connected sets in `ℝⁿ` are measurable (leanprover-community#16976)
move old README.md to OLD_README.md
doc: Add a warning mentioning Lean 4 to the readme (leanprover-community#19243)
feat(topology/metric_space): diameter of pointwise zero and addition (leanprover-community#19028)
feat(topology/algebra/order/liminf_limsup): Eventual boundedness of neighborhoods (leanprover-community#18629)
feat(probability/independence): Independence of singletons (leanprover-community#18506)
feat(combinatorics/set_family/ahlswede_zhang): Ahlswede-Zhang identity, part I (leanprover-community#18612)
feat(data/finset/lattice): `sup'`/`inf'` lemmas (leanprover-community#18989)
chore(order/liminf_limsup): Generalise and move lemmas (leanprover-community#18628)
feat(algebraic_topology/dold_kan): The Dold-Kan equivalence for abelian categories (leanprover-community#17926)
feat(data/sum/interval): The lexicographic sum of two locally finite orders is locally finite (leanprover-community#11352)
feat(analysis/convex/proj_Icc): Extending convex functions (leanprover-community#18797)
feat(algebraic_topology/dold_kan): The Dold-Kan equivalence for pseudoabelian categories (leanprover-community#17925)
feat(measure_theory/measure/haar_quotient): the Unfolding Trick (leanprover-community#18863)
feat(linear_algebra/orientation): add `orientation.reindex` (leanprover-community#19236)
feat(combinatorics/quiver/covering): Definition of coverings and unique lifting of paths (leanprover-community#17828)
feat(set_theory/game/pgame): small sets of pre-games / games / surreals are bounded (leanprover-community#15260)
feat(tactic/positivity): Extension for `ite` (leanprover-community#17650)
...
# Conflicts:
# README.md
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YaelDillies
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Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
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YaelDillies
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Jun 17, 2024
Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
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Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
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AntoineChambert-Loir
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Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
grunweg
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Jun 20, 2024
Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
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Prove that the frontier of an order-connected set in `ℝⁿ` (with the `∞`-metric, but it doesn't actually matter) has measure zero. As a corollary, antichains in `ℝⁿ` have measure zero. This is not so trivial as one might think. The proof Kexing and I came up with involves the Lebesgue density theorem. Partially forward-port leanprover-community/mathlib3#16976
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Prove that the frontier of an order-connected set in
ℝⁿ(with the∞-metric, but it doesn't actually matter) has measure zero.As a corollary, antichains in
ℝⁿhave measure zero.Co-authored-by: @JasonKYi
ennreal.of_real#16955nhds_withinversions #16956a * b ≠ b ↔ a ≠ 1#18635This is a three lines remark in Reverse Kleitman Inequalities, Bollobas, Leader, Radcliffe, with no proof. So Jason, Bhavik and I came up with a proof using Lebesgue Density Theorem.