[Merged by Bors] - refactor(measure_theory/simple_func_dense): generalize approximation results from L^1 to L^p #8114
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Sure, that sounds like a good rule to have! I'll make the change. |
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Co-authored-by: Rémy Degenne <remydegenne@gmail.com>
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Co-authored-by: Rémy Degenne <remydegenne@gmail.com>
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…results from L^1 to L^p (#8114) Simple functions are dense in L^p. The argument for `1 ≤ p < ∞` is exactly the same as for `p = 1`, which was already in mathlib: construct a suitable sequence of pointwise approximations and apply the Dominated Convergence Theorem. This PR refactors to provide the general-`p` result. The argument for `p = ∞` requires finite-dimensionality of `E` and a different approximating sequence, so is left for another PR.
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…construction of embedding of L1 simple funcs (#8185) At the moment, there is a low-level construction in `measure_theory/simple_func_dense` for the approximation of an element of L1 by simple functions, and this is generalized to a more abstract version (with a normed space `L1.simple_func` and a dense linear embedding of this into `L1`) in `measure_theory/bochner_integration`. #8114 generalized the low-level construction, and I am thinking of rewriting the more abstract version to apply to `Lp`, too. But since this will all be more generally useful than in the construction of the Bochner integral, and since the Bochner integral file is very long, I propose moving all this material into `measure_theory/simple_func_dense`. This PR shows what it would look like. There are no mathematical changes.
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…results from L^1 to L^p (#8114) Simple functions are dense in L^p. The argument for `1 ≤ p < ∞` is exactly the same as for `p = 1`, which was already in mathlib: construct a suitable sequence of pointwise approximations and apply the Dominated Convergence Theorem. This PR refactors to provide the general-`p` result. The argument for `p = ∞` requires finite-dimensionality of `E` and a different approximating sequence, so is left for another PR.
b-mehta
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…construction of embedding of L1 simple funcs (#8185) At the moment, there is a low-level construction in `measure_theory/simple_func_dense` for the approximation of an element of L1 by simple functions, and this is generalized to a more abstract version (with a normed space `L1.simple_func` and a dense linear embedding of this into `L1`) in `measure_theory/bochner_integration`. #8114 generalized the low-level construction, and I am thinking of rewriting the more abstract version to apply to `Lp`, too. But since this will all be more generally useful than in the construction of the Bochner integral, and since the Bochner integral file is very long, I propose moving all this material into `measure_theory/simple_func_dense`. This PR shows what it would look like. There are no mathematical changes.
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Simple functions are dense in L^p. The argument for
1 ≤ p < ∞is exactly the same as forp = 1, which was already in mathlib: construct a suitable sequence of pointwise approximations and apply the Dominated Convergence Theorem. This PR refactors to provide the general-presult.The argument for
p = ∞requires finite-dimensionality ofEand a different approximating sequence, so is left for another PR.