[Merged by Bors] - feat(measure_theory/lp_space): add more API on Lp spaces #6042
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Could you please open a PR from https://github.com/leanprover-community/mathlib/tree/split-ae-eq-fun (cherry-picked from your diff) and merge it right away? |
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I golfed a few proofs, feel free to revert (some of) my changes or leave as is. |
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Flesh out the file on `L^p` spaces, notably adding facts on the composition with Lipschitz functions. This makes it possible to define in one go the positive part of an `L^p` function, and its image under a continuous linear map. The file `ae_eq_fun.lean` is split into two, to solve a temporary issue: this file defines a global emetric space instance (of `L^1` type) on the space of function classes. This passes to subtypes, and clashes with the topology on `L^p` coming from the distance. This did not show up before as there were not enough topological statements on `L^p`, but I have been bitten by this when adding new results. For now, we move this part of `ae_eq_fun.lean` to another file which is not imported by `lp_space.lean`, to avoid the clash. This will be solved in a subsequent PR in which I will remove the global instance, and construct the integral based on the specialization of `L^p` to `p = 1` instead of the ad hoc construction we have now (note that, currently, we have two different `L^1` spaces in mathlib, denoted `Lp E 1 μ` and `α →₁[μ] E` -- I will remove the second one in a later PR). Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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Flesh out the file on `L^p` spaces, notably adding facts on the composition with Lipschitz functions. This makes it possible to define in one go the positive part of an `L^p` function, and its image under a continuous linear map. The file `ae_eq_fun.lean` is split into two, to solve a temporary issue: this file defines a global emetric space instance (of `L^1` type) on the space of function classes. This passes to subtypes, and clashes with the topology on `L^p` coming from the distance. This did not show up before as there were not enough topological statements on `L^p`, but I have been bitten by this when adding new results. For now, we move this part of `ae_eq_fun.lean` to another file which is not imported by `lp_space.lean`, to avoid the clash. This will be solved in a subsequent PR in which I will remove the global instance, and construct the integral based on the specialization of `L^p` to `p = 1` instead of the ad hoc construction we have now (note that, currently, we have two different `L^1` spaces in mathlib, denoted `Lp E 1 μ` and `α →₁[μ] E` -- I will remove the second one in a later PR). Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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Flesh out the file on
L^pspaces, notably adding facts on the composition with Lipschitz functions. This makes it possible to define in one go the positive part of anL^pfunction, and its image under a continuous linear map.The file
ae_eq_fun.leanis split into two, to solve a temporary issue: this file defines a global emetric space instance (ofL^1type) on the space of function classes. This passes to subtypes, and clashes with the topology onL^pcoming from the distance. This did not show up before as there were not enough topological statements onL^p, but I have been bitten by this when adding new results. For now, we move this part ofae_eq_fun.leanto another file which is not imported bylp_space.lean, to avoid the clash. This will be solved in a subsequent PR in which I will remove the global instance, and construct the integral based on the specialization ofL^ptop = 1instead of the ad hoc construction we have now (note that, currently, we have two differentL^1spaces in mathlib, denotedLp E 1 μandα →₁[μ] E-- I will remove the second one in a later PR).The only file that is really changed in the PR is
lp_space.lean, the rest is just the splitting ofae_eq_funinto two parts, without changing a line. Inlp_space.lean, I have tried to uniformize things a little bit. In particular,pis now used consistently as an ennreal, andqas a real, as opposed to the previous situation where the roles ofpandqwere exchanged in the beginning of the file, for mainly historical reasons. This makes a big diff, but a lot of it is just this trivial substitution.