Plain-English statement
The Kantorovich cost is the smallest extended expected cost among all couplings of two measures.
Mathematical statement
T_c(mu,nu) = inf_{gamma in C(mu,nu)} integral c(x,y) dgamma(x,y).
Intuition
A coupling chooses how mass from the first law is paired with mass from the second; the infimum selects the cheapest such joint law.
Conditions
- two measurable state spaces and measures
- an ENNReal-valued cost
- the feasible set consists of measures with the prescribed marginals
Why these conditions cannot be dropped
- ENNReal preserves the possibility of infinite cost
- lower semicontinuity is not needed for the value definition but is needed to prove a minimizer exists
Proof route
- define the coupling set through first and second marginals
- take sInf of all coupling lintegral values
- use the product measure to verify nonemptiness for probability marginals
Lean interface notes
- lintegral is used because the cost is ENNReal-valued
- sInf is noncomputable and represents the extended-real infimum
- no optimal transport plan is constructed here
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Lean statement
noncomputable def transportCost {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(c : α × β → ℝ≥0∞) (μ : Measure α) (ν : Measure β) : ℝ≥0∞ :=
sInf {r : ℝ≥0∞ | ∃ γ ∈ couplingSet μ ν, r = ∫⁻ z, c z ∂γ}
/-- Source-facing expansion of the Kantorovich value in display (1.3.2). -/Open AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:36published source at 7bcd37294df1