[Merged by Bors] - feat: generalize ae_hasDerivAt_integral to Banach spaces

[certik]

Generalize both interval-version Lebesgue differentiation theorems LocallyIntegrable.ae_hasDerivAt_integral and
IntervalIntegrable.ae_hasDerivAt_integral from real-valued functions f : ℝ → ℝ to functions f : ℝ → E valued in a Banach space E. The existing proof already goes through the vector-valued averaging theorem VitaliFamily.ae_tendsto_average, so the only change is replacing scalar multiplication * by • in the slope computation.

This is a prerequisite for #40973.

AI usage disclosure: I used Claude Opus 4.8 to implement this and manually tested it with the other PR and my other separate project.

[github-actions]

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[github-actions]

PR summary b459ac2de9

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

No declarations were harmed in the making of this PR! 🐙

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit b459ac2).

• +0 new declarations
• −0 removed declarations

No declaration differences.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit b459ac2de9
Reference commit d9a189b1cc

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[certik]

LLM-generated

[mcdoll]

Thanks!
maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by mcdoll.

[eric-wieser]

bors merge

Thanks!

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Post-CI job
• ci (staging) / Lint style
• ci (staging) / Test and lint
• ci (staging) / Post-Build Step
• ci (staging) / Build

[certik]

Thanks @eric-wieser and @mcdoll !