mathlib documentation

measure_theory.decomposition

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theorem measure_theory.hahn_decomposition {α : Type u_1} [measurable_space α] {μ ν : measure_theory.measure α} (hμ : ⇑μ set.univ < ⊤) (hν : ⇑ν set.univ < ⊤) :
∃ (s : set α), measurable_set s ∧ (∀ (t : set α), measurable_set t → t ⊆ s → ⇑ν t ≤ ⇑μ t) ∧ ∀ (t : set α), measurable_set t → t ⊆ sᶜ → ⇑μ t ≤ ⇑ν t