feat(Analysis/InnerProductSpace/Reproducing): add bilinear form and integral operator for Mercers theorem

[TJHeeringa]

Mercer's theorem is an important theorem for RKHS. The theorem follows from properties of an integral operator. This introduces that operator and the bilinear form used to define it.

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The proof of Mercer's theorem relies on showing that defined integralOperator is a Hilbert-Schmidt operator. These are not part of lean yet, so Mercer's theorem itself cannot be proven yet.

The proof of bilinearity of mercerForm has 4 very similar bits. I suspect that can be shortened, but not figured out how yet.

Still a work in progress because lemmas like integralOperator μ hK f =ᵐ[μ] fun x ↦ ∫ (y : X), K x y (f y) ∂μ are still missing. A list is below. Some should perhaps be PRs in their own right, after this one.

• show integralOperator_apply
• show mercerForm is self-adjoint
• show integralOperator is self-adjoint
• show integralOperator is Hilbert-Schmidt
• move to own file
• split enorm_... into own PR
• ...

In the git history, you can find that a proof of the properties of the integralOperator directly, instead of via the bilinear form mercerForm. mercerForm is there because the self-adjoint property, Hilbert-Schmidt property, etc. are easier to show indirectly via it.

AI:
Claude and Mistral have helped me get up to speed with how to do the measurability proofs.

[github-actions]

PR summary d23b332538

Import changes exceeding 2%

%	File
+6.80%	Mathlib.Analysis.InnerProductSpace.Reproducing

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Analysis.InnerProductSpace.Reproducing	2530	2702	+172 (+6.80%)

Import changes for all files

Files	Import difference
Mathlib.Analysis.InnerProductSpace.Reproducing	172

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Declarations diff (regex)

+ enorm_inner_le_enorm
+ integralOperator
+ integralOperator_inner_right_eq_mercerForm
+ lintegral_norm_inner_le
+ meas_fst
+ meas_snd
+ mercerForm
+ mercerForm_integrable

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 -- pending)

Computed after the build finishes.

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No changes to strong technical debt.

No changes to weak technical debt.

Current commit d23b332538
Reference commit 4608056c77

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).