[Merged by Bors] - feat: integral of a continuous bilinear map applied to independent random variables

[EtienneC30]

If X and Y are independent then ∫ ω, X ω * Y ω ∂μ = ∫ ω, X ω ∂μ * ∫ ω, Y ω ∂μ.

We generalize this to the case of a general continuous bilinear map, and specialize it for scalar multiplication and multiplication.

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• depends on: [Merged by Bors] - feat: the identity function is a.e.-strongly measurable w.r.t. the map by an a.e.-strongly measurable function #39644

[github-actions]

PR summary e6e0db9d6c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ IndepFun.integrable_bilin
+ IndepFun.integrable_left_of_integrable_op
+ IndepFun.integrable_op
+ IndepFun.integrable_right_of_integrable_op
+ IndepFun.integrable_smul
+ IndepFun.integral_bilin
+ IndepFun.integral_bilin'
+ IndepFun.integral_bilin_comp_comp
+ IndepFun.integral_bilin_comp_comp'
+ IndepFun.integral_comp_smul_comp
+ IndepFun.integral_fun_comp_smul_comp
+ IndepFun.integral_fun_smul_eq_smul_integral
+ IndepFun.integral_smul_eq_smul_integral
+ integral_prod_bilin

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.

---

No changes to strong technical debt.

No changes to weak technical debt.

[sgouezel]

The better generalization is the following: if X and Y are independent and integrable, and B is continuous bilinear, then B X Y is integrable, with integral B (\int X) (\int Y). Does your proof also give that?

[EtienneC30]

When I started this I intended to do it for a general bilinear map but I was not satisfied because I could not drop the integrability condition as can be done with multiplication. In the end I proved a version for bilinear maps with the extra-assumption that ∀ x y, c * ‖x‖ * ‖y‖ ≤ ‖B x y‖ for some c > 0. That way things generalize nicely I think.

[sgouezel]

do you really need the second countable topology here? I haven't thought about it, but since X is integrable everything should happen in a second-countable part of the space. Maybe for the proof it's better to prove this one first and then deduce the version with the f?

[EtienneC30]

I opened #39644 which proves that the identity is a.e.-strongly measurable with respect to µ.map f if f is a.e.-strongly measurable.

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat: the identity function is a.e.-strongly measurable w.r.t. the map by an a.e.-strongly measurable function #39644
  By Dependent Issues (🤖). Happy coding!

[sgouezel]

bors r+
Thanks!

[mathlib-bors]

Pull request successfully merged into master.

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