Subtraction of measures #

In this file we define μ - ν to be the least measure τ such that μ ≤ τ + ν. It is the equivalent of (μ - ν) ⊔ 0 if μ and ν were signed measures. Compare with ENNReal.instSub. Specifically, note that if you have α = {1,2}, and μ {1} = 2, μ {2} = 0, and ν {2} = 2, ν {1} = 0, then (μ - ν) {1, 2} = 2. However, if μ ≤ ν, and ν univ ≠ ∞, then (μ - ν) + ν = μ.

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noncomputable instance MeasureTheory.Measure.instSub {α : Type u_1} [MeasurableSpace α] :

Sub (MeasureTheory.Measure α)

The measure μ - ν is defined to be the least measure τ such that μ ≤ τ + ν. It is the equivalent of (μ - ν) ⊔ 0 if μ and ν were signed measures. Compare with ENNReal.instSub. Specifically, note that if you have α = {1,2}, and μ {1} = 2, μ {2} = 0, and ν {2} = 2, ν {1} = 0, then (μ - ν) {1, 2} = 2. However, if μ ≤ ν, and ν univ ≠ ∞, then (μ - ν) + ν = μ.

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MeasureTheory.Measure.instSub = { sub := fun (μ ν : MeasureTheory.Measure α) => sInf {τ : MeasureTheory.Measure α | μ ≤ τ + ν} }

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theorem MeasureTheory.Measure.sub_def {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} :

μ - ν = sInf {d : MeasureTheory.Measure α | μ ≤ d + ν}

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theorem MeasureTheory.Measure.sub_le_of_le_add {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {d : MeasureTheory.Measure α} (h : μ ≤ d + ν) :

μ - ν ≤ d

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theorem MeasureTheory.Measure.sub_eq_zero_of_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (h : μ ≤ ν) :

μ - ν = 0

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theorem MeasureTheory.Measure.sub_le {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} :

μ - ν ≤ μ

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@[simp]

theorem MeasureTheory.Measure.sub_top {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} :

μ - ⊤ = 0

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@[simp]

theorem MeasureTheory.Measure.zero_sub {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} :

0 - μ = 0

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@[simp]

theorem MeasureTheory.Measure.sub_self {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} :

μ - μ = 0

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theorem MeasureTheory.Measure.sub_apply {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsFiniteMeasure ν] (h₁ : MeasurableSet s) (h₂ : ν ≤ μ) :

(μ - ν) s = μ s - ν s

This application lemma only works in special circumstances. Given knowledge of when μ ≤ ν and ν ≤ μ, a more general application lemma can be written.

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