MeasureTheory.Measure.rnDeriv_restrict_restrict
From the authors
The Radon–Nikodym derivative of μ.restrict s with respect to ν.restrict s is the
Radon–Nikodym derivative of μ with respect to ν, ν.restrict s-almost everywhere.
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α : Type u_1mα : MeasurableSpace αA measurable space is a space equipped with a σ-algebra.
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μ : Measure αA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. -
ν : Measure αμ.HaveLebesgueDecomposition νA pair of measuresμandνis said toHaveLebesgueDecompositionif there exists a measureξand a measurable functionf, such thatξis mutually singular with respect toνandμ = ξ + ν…SigmaFinite νA measureμis called σ-finite if there is a countable collection of sets{ A i | i ∈ ℕ }such thatμ (A i) < ∞and⋃ i, A i = s. -
s : Set αA set is a collection of elements of some typeα.
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hs : MeasurableSet sMeasurableSet smeans thatsis measurable (in the ambient measure space onα)
(μ.restrict s).rnDeriv (ν.restrict s) =ᵐ[ν.restrict s] μ.rnDeriv νf and g are *eventually equal* along a filter l if the set of x such that f x = g x belongs to l.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version is written `μ.real s`.
MeasureTheory.Measure.HaveLebesgueDecomposition : {α : Type u_1} → {m : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → PropA pair of measures `μ` and `ν` is said to `HaveLebesgueDecomposition` if there exists a measure `ξ` and a measurable function `f`, such that `ξ` is mutually singular with respect to `ν` and `μ = ξ + ν.withDensity f`.
MeasureTheory.SigmaFinite : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called σ-finite if there is a countable collection of sets
`{ A i | i ∈ ℕ }` such that `μ (A i) < ∞` and `⋃ i, A i = s`.Set : Type u → Type uA set is a collection of elements of some type `α`.
Although `Set` is defined as `α → Prop`, this is an implementation detail which should not be
relied on. Instead, `Set.ofPred` (also written `{x | p x}`) and membership of a set (`∈`) should be
used to convert between sets and predicates.MeasurableSet : {α : Type u_1} → [MeasurableSpace α] → Set α → Prop`MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`)
Filter.EventuallyEq : {α : Type u_1} → {β : Type u_2} → Filter α → (α → β) → (α → β) → PropTwo functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that `f x = g x` belongs to `l`.
MeasureTheory.Measure.restrict : {α : Type u_2} → {_m0 : MeasurableSpace α} → MeasureTheory.Measure α → Set α → MeasureTheory.Measure αRestrict a measure `μ` to a set `s`.
MeasureTheory.Measure.rnDeriv : {α : Type u_2} → {m : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → α → ENNRealIf a pair of measures `HaveLebesgueDecomposition`, then `rnDeriv` chooses the measurable function from `HaveLebesgueDecomposition`, otherwise it returns the zero function. For sigma-finite measures, `μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)`.
Code
lemma rnDeriv_restrict_restrict (μ ν : Measure α) [μ.HaveLebesgueDecomposition ν] [SigmaFinite ν]
{s : Set α} (hs : MeasurableSet s) :
(μ.restrict s).rnDeriv (ν.restrict s) =ᵐ[ν.restrict s] μ.rnDeriv νProof
by
refine (eq_rnDeriv (s := (μ.singularPart ν).restrict s) (measurable_rnDeriv μ ν)
(((mutuallySingular_singularPart μ ν).restrict s).mono le_rfl restrict_le_self) ?_).symm
rw [← restrict_withDensity hs, ← restrict_add, ← haveLebesgueDecomposition_add μ ν]Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.
Audit surface: 0 project declarations, 13 external constants
✓ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.