The bounded-Lipschitz dual discrepancy is bounded by split transport cost

Given: The signed measures and matching Jordan-part masses of the Statement, with finite first moments of their total variations.

Assume the Axiom of Choice for existence of the Jordan parts (The Axiom of Choice).

Jordan decomposition gives positive mutually singular parts ${\omega }^{+},{\omega }^{-}$ and ${\nu }^{+},{\nu }^{-}$ (Jordan decomposition of a signed measure into unique mutually singular positive parts, Measures on sigma-algebras).

The product of finite measures exists with the product marginals, and Tonelli integrates nonnegative costs (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); the product sigma-algebra of Euclidean Borel spaces is the Euclidean product Borel sigma-algebra (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}). The Euclidean triangle inequality gives $\mathrm{\mid }x-y\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation). Also, $\mathrm{\mid }\omega \mathrm{\mid }={\omega }^{+}+{\omega }^{-}$ and $\mathrm{\mid }\nu \mathrm{\mid }={\nu }^{+}+{\nu }^{-}$; hence each Jordan part is dominated by its total variation measure. For a nonnegative simple function, the simple integral is monotone under this measure domination by its defining finite sum; taking the supremum over simple minorants in the nonnegative integral definition gives $\int \mathrm{\mid }x\mathrm{\mid }d{\omega }^{\sigma }\le \int \mathrm{\mid }x\mathrm{\mid }d\mathrm{\mid }\omega \mathrm{\mid }$ and the analogous bound for ${\nu }^{\sigma }$ (For a signed measure, total variation is nu-plus plus nu-minus, finite partitions suffice, and nu-plus and nu-minus are extremal, The nonnegative Lebesgue integral, The integral of a nonnegative simple function).

The definitions of $d_{BL}$ and $C_{\pm }$ use tests bounded by one with Lipschitz constant at most one, and positive/negative couplings (Bounded-Lipschitz dual discrepancy for finite signed measures, Sign-preserving transport cost for finite signed measures).