feat(MeasureTheory.VectorMeasure): variation defined as a supremum is equal to variation defined using the Hahn-Jordan decomposition#26168
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…n for VectorMeasure (#26156) This PR adds variation for any `VectorMeasure` using a supremum definition. Currently mathlib has `TotalVariation` defined for a signed measure using the Hahn-Jordan decomposition, but this doesn't generalise. Motivation: generally this is an important concept but specifically as a step for proving RMK in the complex case which in turn is a step to prove the spectral theorem. This PR was migrated from #25442. PR divided into smaller pieces, this is just the definition without additional lemmas. PRs adding further results related to variation are: * #26160 * #26165 * #26168 (shows that for `SignedMeasures` the two definitions of variation coincide) Co-authored-by: @yoh-tanimoto Co-authored-by: Yoh Tanimoto <[email protected]> Co-authored-by: Yoh Tanimoto <[email protected]>
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…n for VectorMeasure (leanprover-community#26156) This PR adds variation for any `VectorMeasure` using a supremum definition. Currently mathlib has `TotalVariation` defined for a signed measure using the Hahn-Jordan decomposition, but this doesn't generalise. Motivation: generally this is an important concept but specifically as a step for proving RMK in the complex case which in turn is a step to prove the spectral theorem. This PR was migrated from leanprover-community#25442. PR divided into smaller pieces, this is just the definition without additional lemmas. PRs adding further results related to variation are: * leanprover-community#26160 * leanprover-community#26165 * leanprover-community#26168 (shows that for `SignedMeasures` the two definitions of variation coincide) Co-authored-by: @yoh-tanimoto Co-authored-by: Yoh Tanimoto <[email protected]> Co-authored-by: Yoh Tanimoto <[email protected]>
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…n for VectorMeasure (leanprover-community#26156) This PR adds variation for any `VectorMeasure` using a supremum definition. Currently mathlib has `TotalVariation` defined for a signed measure using the Hahn-Jordan decomposition, but this doesn't generalise. Motivation: generally this is an important concept but specifically as a step for proving RMK in the complex case which in turn is a step to prove the spectral theorem. This PR was migrated from leanprover-community#25442. PR divided into smaller pieces, this is just the definition without additional lemmas. PRs adding further results related to variation are: * leanprover-community#26160 * leanprover-community#26165 * leanprover-community#26168 (shows that for `SignedMeasures` the two definitions of variation coincide) Co-authored-by: @yoh-tanimoto Co-authored-by: Yoh Tanimoto <[email protected]> Co-authored-by: Yoh Tanimoto <[email protected]>
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…n for VectorMeasure (leanprover-community#26156) This PR adds variation for any `VectorMeasure` using a supremum definition. Currently mathlib has `TotalVariation` defined for a signed measure using the Hahn-Jordan decomposition, but this doesn't generalise. Motivation: generally this is an important concept but specifically as a step for proving RMK in the complex case which in turn is a step to prove the spectral theorem. This PR was migrated from leanprover-community#25442. PR divided into smaller pieces, this is just the definition without additional lemmas. PRs adding further results related to variation are: * leanprover-community#26160 * leanprover-community#26165 * leanprover-community#26168 (shows that for `SignedMeasures` the two definitions of variation coincide) Co-authored-by: @yoh-tanimoto Co-authored-by: Yoh Tanimoto <[email protected]> Co-authored-by: Yoh Tanimoto <[email protected]>
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signedMeasure_totalVariation_eq: ifμis aSignedMeasurethen variation defined as a supremum is equal to variation defined using the Hahn-Jordan decomposition.Co-authored-by: @yoh-tanimoto
ℝ≥0∞VectorMeasure is equal to itself #26165