[Merged by Bors] - feat: induction principles for strongly measurable functions #22960
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PR summary d7487c74b4Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Introduce two induction principles for strongly measurable functions. To prove a predicate for all strongly measurable functions it is enough to show that it holds for simple functions and that it is closed under pointwise limits. To prove a property for simple functions, we can prove that it is true for constant indicators and closed under addition if the codomain is an [AddZeroClass](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Defs.html#AddZeroClass) (this is [MeasureTheory.SimpleFunc.induction](https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SimpleFunc.html#MeasureTheory.SimpleFunc.induction)). If the codomain is not an `AddZeroClass`, we can prove a property on simple functions by showing that it holds for constant functions and is closed under piecewise combinations of functions. We introduce an induction principle for simple functions following this idea, as well as a corresponding version for strongly measurable functions.
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Introduce two induction principles for strongly measurable functions. To prove a predicate for all strongly measurable functions it is enough to show that it holds for simple functions and that it is closed under pointwise limits.
To prove a property for simple functions, we can prove that it is true for constant indicators and closed under addition if the codomain is an AddZeroClass (this is MeasureTheory.SimpleFunc.induction).
If the codomain is not an
AddZeroClass, we can prove a property on simple functions by showing that it holds for constant functions and is closed under piecewise combinations of functions. We introduce an induction principle for simple functions following this idea, as well as a corresponding version for strongly measurable functions.Co-authored-by: @D-Thomine