ProbabilityMeasure.lean`

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Annotation provenance. This overlay contains 9 curated AI-written annotations and 485 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Polynomial Freiman-Ruzsa proof paper, arXiv v2

9 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Imported by 3 files

instance (instance at line 13) line 13 source
Source documentation
A Measure which is a probability measure can be lifted to a ProbabilityMeasure.
```
instance {Y : Type*} [MeasurableSpace Y] :
    CanLift (Measure Y) (ProbabilityMeasure Y) (↑) IsProbabilityMeasure where
…
```
instance (instance at line 18) line 18 source
Source documentation
A Measure which is a finite measure can be lifted to a FiniteMeasure.
```
instance {Y : Type*} [MeasurableSpace Y] :
    CanLift (Measure Y) (FiniteMeasure Y) (↑) IsFiniteMeasure where
…
```
lemma continuous_finiteMeasure_apply_of_isClopen The measure of any connected component depends continuously on the FiniteMeasure line 25 source
Source documentation
The measure of any connected component depends continuously on the FiniteMeasure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, OpensMeasurableSpace, Set, IsClopen, Continuous, MeasureTheory.FiniteMeasure, Real, MeasureTheory.Measure.real
Proof uses: Eq.mpr, MeasureTheory.integral, BoundedContinuousFunction, BoundedContinuousFunction.indicator, eq_of_heq, Eq, HEq, Eq.refl, HEq.refl, Eq.symm, funext, MeasurableSet, IsOpen.measurableSet, IsClopen.isOpen, of_eq_true, ENNReal.toReal, MeasureTheory.Measure, ENNReal, Eq.trans, True and 16 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma continuous_finiteMeasure_apply_of_isClopen [OpensMeasurableSpace X]
    {s : Set X} (s_clopen : IsClopen s) :
    Continuous fun μ : FiniteMeasure X ↦ (μ : Measure X).real s := by
…
```
lemma continuous_probabilityMeasure_apply_of_isClopen The probability of any connected component depends continuously on the ProbabilityMeasure line 34 source
Source documentation
The probability of any connected component depends continuously on the ProbabilityMeasure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, OpensMeasurableSpace, Set, IsClopen, Continuous, MeasureTheory.ProbabilityMeasure, Real, MeasureTheory.Measure.real
Proof uses: Eq.mpr, MeasureTheory.integral, BoundedContinuousFunction, BoundedContinuousFunction.indicator, eq_of_heq, Eq, HEq, Eq.refl, HEq.refl, Eq.symm, funext, MeasurableSet, IsOpen.measurableSet, IsClopen.isOpen, of_eq_true, ENNReal.toReal, MeasureTheory.Measure, ENNReal, Eq.trans, True and 16 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma continuous_probabilityMeasure_apply_of_isClopen [OpensMeasurableSpace X]
    {s : Set X} (s_clopen : IsClopen s) :
    Continuous fun μ : ProbabilityMeasure X ↦ (μ : Measure X).real s := by
…
```
lemma continuous_pmf_apply' line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, DiscreteTopology, BorelSpace, Continuous, MeasureTheory.ProbabilityMeasure, Real, MeasureTheory.Measure.real, Singleton.singleton, Set
Proof uses: continuous_probabilityMeasure_apply_of_isClopen, isClopen_discrete
9 instances, coercions and auxiliary constants left out.
lemma continuous_pmf_apply line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, DiscreteTopology, BorelSpace, Continuous, MeasureTheory.ProbabilityMeasure, NNReal, Set, Singleton.singleton
Proof uses: Eq.mpr, Function.comp, Real, Real.toNNReal, MeasureTheory.Measure.real, eq_of_heq, Eq, HEq, Eq.refl, HEq.refl, Eq.symm, funext, NNReal.eq, ENNReal.toNNReal, MeasureTheory.Measure, ENNReal, id, Eq.trans, congrArg, ENNReal.toNNReal_toReal_eq and 3 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma tendsto_lintegral_of_forall_of_finite line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, DiscreteTopology, BorelSpace, Finite, Filter, MeasureTheory.Measure, BoundedContinuousFunction, NNReal, Filter.Tendsto, ENNReal, Set, Singleton.singleton, nhds, MeasureTheory.lintegral
Proof uses: Fintype, Nonempty, Eq, nonempty_fintype, Nonempty.intro, Eq.mpr, Finset.sum, Finset.univ, HMul.hMul, id, congr, congrFun', congrArg, funext, MeasureTheory.lintegral_fintype, tendsto_finsetSum, Membership.mem, Finset, ENNReal.Tendsto.const_mul, Or.inr and 5 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma ProbabilityMeasure.tendsto_iff_forall_apply_tendsto Probability measures on a finite space tend to a limit if and only if the probability masses of all points tend to the corresponding limits line 66 source
Source documentation
Probability measures on a finite space tend to a limit if and only if the probability masses of all points tend to the corresponding limits. Version in ℝ≥0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, DiscreteTopology, BorelSpace, Filter, Finite, MeasureTheory.ProbabilityMeasure, Iff, Filter.Tendsto, nhds, NNReal, Set, Singleton.singleton
Proof uses: Iff.intro, Filter.Tendsto.comp, ContinuousAt.tendsto, Continuous.continuousAt, continuous_pmf_apply, Iff.mpr, BoundedContinuousFunction, ENNReal, MeasureTheory.lintegral, MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto, tendsto_lintegral_of_forall_of_finite, Eq.mpr, MeasureTheory.Measure, Function.comp, eq_of_heq, Eq, HEq, Eq.refl, HEq.refl, Eq.symm and 8 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
lemma ProbabilityMeasure.tendsto_iff_forall_apply_tendsto {L : Filter ι} [Finite X]
    (μs : ι → ProbabilityMeasure X) (μ : ProbabilityMeasure X) :
    Tendsto μs L (𝓝 μ) ↔ ∀ a, Tendsto (μs · {a}) L (𝓝 (μ {a})) := by
…
```
lemma ProbabilityMeasure.tendsto_iff_forall_apply_tendsto_ennreal Probability measures on a finite space tend to a limit if and only if the probability masses of all points tend to the corresponding limits line 82 source
Source documentation
Probability measures on a finite space tend to a limit if and only if the probability masses of all points tend to the corresponding limits. Version in ℝ≥0∞.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, DiscreteTopology, BorelSpace, Filter, Finite, MeasureTheory.ProbabilityMeasure, Iff, Filter.Tendsto, nhds, ENNReal, MeasureTheory.Measure, Set, Singleton.singleton
Proof uses: Eq.mpr, NNReal, id, Eq, congrArg, propext, ProbabilityMeasure.tendsto_iff_forall_apply_tendsto, of_eq_true, Eq.trans, True, forall_congr, congr, congrFun', funext, iff_self
19 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma ProbabilityMeasure.tendsto_iff_forall_apply_tendsto_ennreal {L : Filter ι} [Finite X]
    (μs : ι → ProbabilityMeasure X) (μ : ProbabilityMeasure X) :
    Tendsto μs L (𝓝 μ) ↔ ∀ a, Tendsto (fun n ↦ (μs n : Measure X) {a}) L
      (𝓝 ((μ : Measure X) {a})) := by
…
```