feat: results on inner regularity of finite measures

[JakobStiefel]

in a PseudoEMetricSpace and CompleteSpace with SecondCountableTopology, any finite measure is inner regular with respect to compact sets. In other words, finite measures are tight. Important result in probability

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[github-actions]

PR summary d541b6c223

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.MeasureTheory.Measure.RegularityCompacts (new file)	1365

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Declarations diff

+ InnerRegularCompactLTTop_of_pseudoEMetricSpace_completeSpace_secondCountable
+ InnerRegular_of_pseudoEMetricSpace_completeSpace_secondCountable
+ PolishSpace.innerRegular_isCompact_isClosed_measurableSet
+ exists_isCompact_closure_measure_compl_lt
+ exists_measure_iInter_lt
+ innerRegularWRT_isCompact
+ innerRegularWRT_isCompact_closure
+ innerRegularWRT_isCompact_closure_iff
+ innerRegularWRT_isCompact_closure_of_univ
+ innerRegularWRT_isCompact_isClosed
+ innerRegularWRT_isCompact_isClosed_iff
+ innerRegularWRT_isCompact_isClosed_iff_innerRegularWRT_isCompact_closure
+ innerRegularWRT_isCompact_isClosed_isOpen
+ innerRegularWRT_isCompact_isOpen
+ innerRegularWRT_of_exists_compl_lt
+ innerRegular_isCompact_isClosed_measurableSet_of_finite
+ interUnionBalls
+ isCompact_closure_interUnionBalls
+ tendsto_atTop_zero_iff_le_of_antitone
+ tendsto_atTop_zero_iff_lt_of_antitone
+ totallyBounded_interUnionBalls

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).

[sgouezel]

The PR text is a little bit misleading, as it's not true in general that a finite measure is inner regular wrt compact sets in an emetric space: you need completeness and second countability assumptions (i.e., a polish space), right? Can you adjust the PR text, and the docstring in the file?

I'm also a little bit confused by the fact that the author of the PR is not mentioned as one of the authors of the main file. Can you explain what is going on here?

And thanks for PRing this, this is an important result that should definitely be in mathlib!

[JakobStiefel]

I'm also a little bit confused by the fact that the author of the PR is not mentioned as one of the authors of the main file. Can you explain what is going on here?

The file is originally fom https://github.com/RemyDegenne/kolmogorov_extension4, which isn't a pull request yet. I copied the needed file with the permission of Peter Pfaffelhuber in order to proceed the work on separation of finite measures, #19782.

[sgouezel]

Suggested change

	satisfying `p` exploit the space arbitrarily well, then `μ` is inner regular with respect to
	satisfying `p` cover the space arbitrarily well, then `μ` is inner regular with respect to

[sgouezel]

This one should be an instance. For the main properties (Regular, InnerRegular, InnerRegularCompactLTTop), registering these as instances makes sure that typeclass inference will find it for you for free. For instance, once you register this one as an instance, then Mathlib also knows automatically that such a measure if regular!

It would also be a good idea to register the corresponding instance on a polish space (because the current one won't apply, as a polish space is not a PseudoEmetricSpace).

[sgouezel]

This one should also be an instance. And please also register the Polish space version as an instance.