feat(Probability/Independence): grouping an independent family by pairwise disjoint index sets

[martinamaggio]

If m : ι → MeasurableSpace Ω is an independent family of σ-algebras and G : ι' → Set ι is a
pairwise disjoint family of index sets, then the family of grouped σ-algebras
fun i' ↦ ⨆ j ∈ G i', m j is again independent.

This is the indexed-family version of the existing two-group lemma indep_iSup_of_disjoint, from
which it follows by induction on the finite subfamily, peeling off one group at a time against the
union of the remaining ones. As usual it is proven for the kernel notion of independence
(Kernel.iIndep_iSup_of_disjoint) and specialized to independence w.r.t. a measure
(iIndep_iSup_of_disjoint) and to conditional independence (iCondIndep_iSup_of_disjoint).

The lemma is the natural bridge from an independent family to block/grouping arguments — e.g.
Markov-style properties of recursions driven by disjoint sets of randomness sources. It was
developed for a formalization of end-to-end latency distributions of periodic task chains, where it
establishes that successive chain states, being measurable w.r.t. σ-algebras of disjoint groups of
an independent family, form an independent family themselves.

[github-actions]

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[github-actions]

PR summary 5c4f635b2e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ iCondIndep_iSup_of_disjoint
++ iIndep_iSup_of_disjoint

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 5c4f635).

• +3 new declarations
• −0 removed declarations

+ProbabilityTheory.Kernel.iIndep_iSup_of_disjoint
+ProbabilityTheory.iCondIndep_iSup_of_disjoint
+ProbabilityTheory.iIndep_iSup_of_disjoint

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 5c4f635b2e
Reference commit d0f60b5007

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).