feat: convergence in probability implies convergence in distribution

[RemyDegenne]

Define convergence of distributions of random variables and prove that convergence in probability implies convergence in distribution, as well as Slutsky's theorem on the convergence of a product of random variables (since those two facts follow from the same lemma).

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• depends on: [Merged by Bors] - feat: thickenedIndicator is Lipschitz #30346
• depends on: [Merged by Bors] - feat: add NullMeasurableSet version of setIntegral_mono_on #30385
• depends on: feat: add an implication in the Portmanteau theorem #30402
• depends on: feat: convergence in distribution and continuous mapping theorem #30540
• depends on: [Merged by Bors] - feat: AEMeasurable.dist #30585

[github-actions]

PR summary 7065ce74ef

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.MeasureTheory.Function.ConvergenceInDistribution (new file)	2316

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Declarations diff

+ TendstoInDistribution
+ TendstoInDistribution.add_of_tendstoInMeasure_const
+ TendstoInDistribution.continuous_comp
+ TendstoInDistribution.continuous_comp_prodMk_of_tendstoInMeasure_const
+ TendstoInDistribution.prodMk_of_tendstoInMeasure_const
+ TendstoInMeasure.tendstoInDistribution
+ setIntegral_mono_on'
+ tendstoInDistribution_const
+ tendstoInDistribution_def
+ tendstoInDistribution_of_not_aemeasurable_left
+ tendstoInDistribution_of_not_aemeasurable_right
+ tendstoInDistribution_of_tendstoInMeasure_sub
+ tendstoInDistribution_unique
+ tendsto_iff_forall_lipschitz_integral_tendsto
+ tendsto_integral_thickenedIndicator_of_isClosed
+ tendsto_of_forall_isOpen_le_liminf'
+ tendsto_of_forall_isOpen_le_liminf_nat'
+ tendsto_of_limsup_measure_closed_le
+ tendsto_of_limsup_measure_closed_le'
+ tendsto_of_limsup_measure_closed_le_nat
++ zero_iff

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.

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No changes to technical debt.

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:

• [Merged by Bors] - feat: thickenedIndicator is Lipschitz #30346
• [Merged by Bors] - feat: add NullMeasurableSet version of setIntegral_mono_on #30385
• feat: add an implication in the Portmanteau theorem #30402
• feat: convergence in distribution and continuous mapping theorem #30540
• [Merged by Bors] - feat: AEMeasurable.dist #30585
  By Dependent Issues (🤖). Happy coding!