feat(Dynamics/BirkhoffSum): add the maximal ergodic theorem

[lua-vr]

---

This file exports three constants:

public def birkhoffAverageSupSet (f : α → α) (g : α → ℝ) (a : ℝ) : Set α :=
  {x | ∃ n : ℕ, a < birkhoffAverage ℝ f g n x}

public theorem meas_birkhoffAverageSupSet_smul_const_le_integral (a : ℝ) (ha : 0 < a) :
    μ.real (birkhoffAverageSupSet f g a) • a ≤ ∫ x in birkhoffAverageSupSet f g a, g x ∂μ

public theorem meas_birkhoffAverageSupSet_smul_const_le_norm (a : ℝ) (ha : 0 < a) :
    μ.real (birkhoffAverageSupSet f (fun x ↦ ‖g x‖) a) • a ≤ ∫ x, ‖g x‖ ∂μ

In my interpretation, the other constants are auxiliary, so I did not make them public. If it's of independent interest, birkhoffMax could be exported or moved to a dedicated file, but I'm not sure of that at the moment.

• depends on: feat(Integral/Bochnet/Set): add setIntegral_compl₀ #34030
• depends on: feat(Order/PartialSups): add exists_partialSups_eq #34029
• depends on: feat(Integral.Bochner.Set): add tendsto_setIntegral_of_monotone₀ #34025

[github-actions]

PR summary 721b21cf2b

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Dynamics.BirkhoffSum.Maximal (new file)	2280

---

Declarations diff

+ birkhoffAverageSupSet_eq_birkhoffSupSet
+ birkhoffAverage_iff_birkhoffSum
+ birkhoffMax
+ birkhoffMax_aestronglyMeasurable
+ birkhoffMax_comp_le_succ
+ birkhoffMax_integrable
+ birkhoffMax_integral_le
+ birkhoffMax_le_birkhoffMax
+ birkhoffMax_measurable
+ birkhoffMax_nonneg
+ birkhoffMax_pos_of_mem_support
+ birkhoffMax_succ
+ birkhoffMax_succ'
+ birkhoffSum_aestronglyMeasurable
+ birkhoffSum_integrable
+ birkhoffSum_measurable
+ birkhoffSupSet
+ birkhoffSupSet_eq_iSup_birkhoffMax_support
+ setIntegral_nonneg_on_birkhoffMax_support
+ setIntegral_nonneg_on_birkhoffSupSet
+ tendsto_setIntegral_on_birkhoffMax_support_birkhoffSupSet

You can run this locally as follows

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

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

Increase in tech debt: (relative, absolute) = (3.00, 0.00)

Current number	Change	Type
638	3	erw

Current commit 5f094262c2
Reference commit 721b21cf2b

You can run this locally as

./scripts/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).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(Integral/Bochnet/Set): add setIntegral_compl₀ #34030
• feat(Order/PartialSups): add exists_partialSups_eq #34029
• feat(Integral.Bochner.Set): add tendsto_setIntegral_of_monotone₀ #34025
  By Dependent Issues (🤖). Happy coding!

[lua-vr]

Where do these lemmas belong? I don't want to add MeasureTheory and Integrable to the imports of BirkhoffSum.Basic...