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[Merged by Bors] - feat(integration): integral commutes with continuous linear maps #4167
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from the sphere eversion project.
PatrickMassot
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src/measure_theory/set_integral.lean
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| by simp [l1.clm_apply] | ||
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| /-- Composing `φ : α →₁[μ] E` with `L : E →L[ℝ] F`, seen as a `ℝ`-linear map on `α →₁[μ] E`. -/ | ||
| def l1.clm_applyₗ (L : E →L[ℝ] F) : (α →₁[μ] E) →ₗ[ℝ] (α →₁[μ] F) := |
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It feels strange to have this one in the namespace measure_theory.l1. I'd rather have it in continuous_linear_map, just like continuous_linear_map.l1_apply (maybe with more similar names)
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I changed all names and namespaces. I'm sorry I opened this PR without doing the name cleanup. I wanted Yury to have it and I was too fast.
src/measure_theory/set_integral.lean
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| @@ -464,6 +464,135 @@ lemma continuous_at.integral_sub_linear_is_o_ae | |||
| is_o (λ s, ∫ x in s, f x ∂μ - (μ s).to_real • f a) (λ s, (μ s).to_real) ((𝓝 a).lift' powerset) := | |||
| (ha.mono_left inf_le_left).integral_sub_linear_is_o_ae hfm (μ.finite_at_nhds a) | |||
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| section continuous_linear_map | |||
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section docstring, explaining what you are going to do in this paragraph, and why?
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bors d+ |
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✌️ PatrickMassot can now approve this pull request. To approve and merge a pull request, simply reply with |
Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>
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bors merge |
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👎 Rejected by label |
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from the sphere eversion project. Main result:
```lean
continuous_linear_map.integral_apply_comm {α : Type*} [measurable_space α] {μ : measure α}
{E : Type*} [normed_group E] [second_countable_topology E] [normed_space ℝ E] [complete_space E]
[measurable_space E] [borel_space E] {F : Type*} [normed_group F]
[second_countable_topology F] [normed_space ℝ F] [complete_space F]
[measurable_space F] [borel_space F]
{φ : α → E} (L : E →L[ℝ] F) (φ_meas : measurable φ) (φ_int : integrable φ μ) :
∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ)
```
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Pull request successfully merged into master. Build succeeded: |
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from the sphere eversion project. Main result:
continuous_linear_map.integral_apply_comm {α : Type*} [measurable_space α] {μ : measure α} {E : Type*} [normed_group E] [second_countable_topology E] [normed_space ℝ E] [complete_space E] [measurable_space E] [borel_space E] {F : Type*} [normed_group F] [second_countable_topology F] [normed_space ℝ F] [complete_space F] [measurable_space F] [borel_space F] {φ : α → E} (L : E →L[ℝ] F) (φ_meas : measurable φ) (φ_int : integrable φ μ) : ∫ a, L (φ a) ∂μ = L (∫ a, φ a ∂μ)This PR doesn't build on #4170 but it will conflict, so should be merged later.