transportCost_eq_sInf · Samplinglib

Plain-English statement

The source display is the direct expansion of the Kantorovich transport-cost definition.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

T_c(mu,nu) equals the sInf of integral c dgamma over gamma in the coupling set.

Intuition

This theorem provides a stable source-facing rewrite while keeping existence of a minimizer separate.

Conditions

the same measurable spaces, measures, and ENNReal cost as transportCost

Why these conditions cannot be dropped

the theorem is definitional and therefore introduces no hidden regularity or attainment hypothesis

Proof route

unfold transportCost

Lean interface notes

the proof is rfl
definitional equality does not imply the infimum is attained

Lean learning studio · mathematics → formal proof

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Lean statement

theorem transportCost_eq_sInf
    {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
    (c : α × β → ℝ≥0∞) (μ : Measure α) (ν : Measure β) :
    transportCost c μ ν =
      sInf {r : ℝ≥0∞ | ∃ γ ∈ couplingSet μ ν, r = ∫⁻ z, c z ∂γ} :=
  rfl

/-- A prescribed probability marginal forces the joint coupling measure to
have total mass one. This recovers the probability-measure interface required
by expectations and transport costs from the marginal contract. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:41published source at 7bcd37294df1