Plain-English statement
P2,ac is the class of probability laws with a finite squared-distance moment and a Lebesgue density.
Mathematical statement
mu is a probability measure, mu is absolutely continuous with respect to volume, and integral ||x||^2 dmu is finite.
Intuition
The second-moment condition places the law in quadratic Wasserstein space; absolute continuity is the regularity used later for Brenier maps and Wasserstein calculus.
Conditions
- a finite-dimensional real inner-product space
- the Borel measurable structure and Lebesgue volume
- probability normalization, absolute continuity, and squared-norm integrability
Why these conditions cannot be dropped
- finite second moment is the domain condition for W2
- absolute continuity is needed for source results that use almost-everywhere gradients of convex potentials
- neither condition alone provides an optimal map
Proof route
- package the three source conditions as a reusable predicate
- expose an iff expansion for downstream elaboration
Lean interface notes
- volume is Mathlib's finite-dimensional Lebesgue measure
- Integrable of the nonnegative real squared norm expresses a finite second moment
- the probability assertion remains explicit inside the predicate
Read the mathematics first, then descend into Lean
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Lean statement
def IsAbsolutelyContinuousFiniteSecondMoment
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(μ : Measure E) : Prop :=
IsProbabilityMeasure μ ∧
μ ≪ (volume : Measure E) ∧
Integrable (fun x : E => ‖x‖ ^ 2) μ
/-- Expansion of the three conditions in the `P₂,ac` definition. -/Open AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:49published source at 7bcd37294df1