feat(MeasureTheory): define Hellinger affinity

[sadasant]

Define the Hellinger affinity (Bhattacharyya coefficient) of two measures as an ENNReal-valued lintegral against the canonical dominating measure μ + ν, making the definition total with no absolute-continuity or integrability side conditions. Provide symmetry, invariance under the choice of sigma-finite dominating measure, the self-affinity of a probability measure, the rnDeriv and withDensity computation rules, the bound ≤ 1 for probability measures (via Hölder with p = q = 2), and the characterization that the affinity vanishes iff the measures are mutually singular. Groundwork for a formalization of Kakutani's dichotomy (1948) for infinite products of probability measures.

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Design notes.

On the value type ("why not ℝ / ℝ≥0 / EReal"): the ℝ≥0∞-valued lintegral against μ + ν makes the affinity total, which is what makes the singular direction of Kakutani's dichotomy (planned follow-up PRs) case-split-free — hellingerAffinity_eq_zero_iff already characterizes mutual singularity with no absolute-continuity hypothesis. InformationTheory.klDiv sets the precedent of an ℝ≥0∞-valued, standalone divergence-like quantity. The invariance lemma hellingerAffinity_eq_lintegral_rnDeriv_mul_rnDeriv recovers the textbook ∫ √(dμ/dρ · dν/dρ) dρ for any σ-finite dominating ρ.

Relation to f-divergences (CC @RemyDegenne for coordination with the planned fDiv upstreaming from TestingLowerBounds): the affinity is the f-integral at f = √·, equivalently 1 − the Hellinger-½ divergence, with H²(μ,ν)/2 = 1 − affinity. Rather than block on an unlanded framework, this PR keeps the affinity standalone (the klDiv pattern); when fDiv lands, a single bridging lemma of the shape fDiv hellingerFun μ ν = 1 - hellingerAffinity μ ν reconciles them, and I am happy to contribute it. Also happy to move the file to Mathlib/InformationTheory/ if reviewers prefer that home over Mathlib/MeasureTheory/Measure/.

Roadmap. This is PR 1 of a planned sequence building toward Kakutani's dichotomy for Measure.infinitePi over an arbitrary index type (stronger than the textbook countable statement): the lintegral product Fubini and Measure.pi_withDensity, the infinite-product/summability bridges, then the singular and absolutely-continuous directions and the packaged dichotomy. Natural follow-up API for this file, available on request: hellingerAffinity_pos_iff, monotonicity in each argument, and the relation to the squared Hellinger distance once a distance exists.

Attribution and AI disclosure. This material was developed in the public riemann-venue repository by Daniel Rodriguez in collaboration with Claude Fable 5 (Anthropic), as part of a machine-checked formalization of Kakutani's dichotomy (1948) for infinite products of probability measures. The definitions, proofs, and the drafting of this description were done with Claude Fable 5 via Claude Code, working from a design and statement plan reviewed by the human author; all proofs are checked by Lean (#print axioms on the main results reports only propext, Classical.choice, Quot.sound), and the human author reviewed the final form, takes responsibility for it, and will answer all review comments personally. I am applying the LLM-generated label per the contribution guidelines.

[github-actions]

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[github-actions]

PR summary 029c16d920

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.MeasureTheory.Measure.Hellinger (new file)	2353

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Declarations diff (regex)

+ hellingerAffinity
+ hellingerAffinity_comm
+ hellingerAffinity_eq_lintegral_rnDeriv
+ hellingerAffinity_eq_lintegral_rnDeriv_mul_rnDeriv
+ hellingerAffinity_eq_zero_iff
+ hellingerAffinity_le_one
+ hellingerAffinity_self
+ hellingerAffinity_withDensity
+ mul_self_rpow_half

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 029c16d).

• +8 new declarations
• −0 removed declarations

+MeasureTheory.Measure.hellingerAffinity
+MeasureTheory.Measure.hellingerAffinity_comm
+MeasureTheory.Measure.hellingerAffinity_eq_lintegral_rnDeriv
+MeasureTheory.Measure.hellingerAffinity_eq_lintegral_rnDeriv_mul_rnDeriv
+MeasureTheory.Measure.hellingerAffinity_eq_zero_iff
+MeasureTheory.Measure.hellingerAffinity_le_one
+MeasureTheory.Measure.hellingerAffinity_self
+MeasureTheory.Measure.hellingerAffinity_withDensity

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5008	1	exposed public sections

Current commit 029c16d920
Reference commit d45663d02f

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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[sadasant]

LLM-generated