# Documentation

Mathlib.MeasureTheory.Measure.Sub

# 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.

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

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.

theorem MeasureTheory.Measure.sub_def {α : Type u_1} {m : } {μ : } {ν : } :
μ - ν = sInf {d | μ d + ν}
theorem MeasureTheory.Measure.sub_le_of_le_add {α : Type u_1} {m : } {μ : } {ν : } {d : } (h : μ d + ν) :
μ - ν d
theorem MeasureTheory.Measure.sub_eq_zero_of_le {α : Type u_1} {m : } {μ : } {ν : } (h : μ ν) :
μ - ν = 0
theorem MeasureTheory.Measure.sub_le {α : Type u_1} {m : } {μ : } {ν : } :
μ - ν μ
@[simp]
theorem MeasureTheory.Measure.sub_top {α : Type u_1} {m : } {μ : } :
μ - = 0
@[simp]
theorem MeasureTheory.Measure.zero_sub {α : Type u_1} {m : } {μ : } :
0 - μ = 0
@[simp]
theorem MeasureTheory.Measure.sub_self {α : Type u_1} {m : } {μ : } :
μ - μ = 0
theorem MeasureTheory.Measure.sub_apply {α : Type u_1} {m : } {μ : } {ν : } {s : Set α} (h₁ : ) (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.

theorem MeasureTheory.Measure.sub_add_cancel_of_le {α : Type u_1} {m : } {μ : } {ν : } (h₁ : ν μ) :
μ - ν + ν = μ
theorem MeasureTheory.Measure.restrict_sub_eq_restrict_sub_restrict {α : Type u_1} {m : } {μ : } {ν : } {s : Set α} (h_meas_s : ) :
theorem MeasureTheory.Measure.sub_apply_eq_zero_of_restrict_le_restrict {α : Type u_1} {m : } {μ : } {ν : } {s : Set α} (h_le : ) (h_meas_s : ) :
↑(μ - ν) s = 0
instance MeasureTheory.Measure.isFiniteMeasure_sub {α : Type u_1} {m : } {μ : } {ν : } :