feat(MeasureTheory): add IntegrableOn. tendsto_primitive_ over Ioi/Iio and IntegrableOn.continuousOn_primitive_ over Ioi/Iio/Ici/Iic

[Deep0Thinking]

This PR mainly proves:

• IntegrableOn.tendsto_primitive_ over Ioi/Iio
• IntegrableOn.continuousWithinAt_primitive_ over Ioi/Iio
• IntegrableOn.continuousOn_primitive_ over Ioi/Iio/Ici/Iic
• IntegrableOn.tendsto_integral_ over Ioi/Iio/Ici/Iic

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[github-actions]

PR summary a786f33605

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ IntegrableOn.continuousOn_primitive_Ici
+ IntegrableOn.continuousOn_primitive_Iic
+ IntegrableOn.continuousOn_primitive_Iio
+ IntegrableOn.continuousOn_primitive_Ioi
+ IntegrableOn.continuousWithinAt_primitive_Iio
+ IntegrableOn.continuousWithinAt_primitive_Ioi
+ IntegrableOn.tendsto_integral_Ici
+ IntegrableOn.tendsto_integral_Iic
+ IntegrableOn.tendsto_integral_Iio
+ IntegrableOn.tendsto_integral_Ioi
+ IntegrableOn.tendsto_primitive_Iio
+ IntegrableOn.tendsto_primitive_Ioi
+ Ioi_diff_Ioc

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.

---

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).

[Deep0Thinking]

t-measure-probability

[CoolRmal]

Actually I think you should prove the following analogue of intervalIntegral.continuousOn_primitive_Icc first: fun b ↦ ∫ x in Ioi b, f x ∂μ is a right continuous function on [a, ∞).

import Mathlib

open MeasureTheory Set

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : Measure ℝ} {f : ℝ → E}

theorem IntegrableOn.continuousOnIci_primitive {a : ℝ} (hf : IntegrableOn f (Ioi a) μ) :
    ContinuousOn (fun b => ∫ x in Ioi b, f x ∂μ) (Ici a) := by
  sorry

Note that you should also prove this theorem for an arbitrary measure μ. After you proved this, IntegrableOn.tendsto_integral_Ioi should be an easy corollary.

Also why don't you also include the version for Iio, that is, fun b ↦ ∫ x in Iio b, f x ∂μ is left continuous.

If you also want to include the version for Ici or Iic, then you probably need some extra assumptions on the measure μ(e.g. the measure has no atom). Ignore this if you are not very familiar with measure theory.

[Deep0Thinking]

WIP

[Deep0Thinking]

-WIP

[Deep0Thinking]

@CoolRmal Thank you so much for the suggestions!! I've added the theorems of IntegrableOn.continuousOn_primitive_ over Ioi/Iio/Ici/Iic and IntegrableOn.tendsto_primitive_ over Ioi/Iio, with IntegrableOn.tendsto_integral_ over Ioi/Iio/Ici/Iic as corollaries.