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A finite coupling lifts to a Borel coupling with a controlled metric cost

Statement

Let $(X, d)$ be a metric space that is compact or second-countable. Let $x_{1}, \dots, x_{m}$ be distinct points, let $C_{1}, \dots, C_{m}$ be a Borel partition of $X$ , and let $ε > 0$ satisfy $d (x, x_{i}) < ε$ for every $x \in C_{i}$ . Let $μ, ν$ be Borel probability measures on $X$ , set $p_{i} = μ (C_{i})$ and $q_{j} = ν (C_{j})$ , and let $π = (π_{i j})$ be a finite coupling of the probability vectors $p, q$ . Then there is a Borel probability measure $γ$ on $X \times X$ with marginals $μ, ν$ such that $\int_{X \times X} d (x, y) d γ (x, y) \leq \sum_{i, j = 1}^{m} π_{i j} d (x_{i}, x_{j}) + 2 ε .$

Facts & Assumptions

Given: The space, partition, measures, and finite coupling in the Statement.

[F1]

For second-countable spaces with product topology, the Borel sigma-algebra on $X \times X$ equals $B (X) \otimes B (X)$ . Borel sigma-algebra of a product of second-countable spaces

[F2]

For sigma-finite factors the product measure exists on the product sigma-algebra and has the product rectangle values. For Borel probabilities on compact metric spaces, the section construction gives a probability on the full product-topology Borel sigma-algebra with the same rectangle values, without second countability. For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique Borel product probabilities on compact metric spaces by sections Open cover, subcover, compact metric space, and compact subset of a metric space

[F3]

A finite coupling has nonnegative entries and the prescribed row and column sums. Couplings of finite probability vectors

[F4]

A probability vector has nonnegative coordinates summing to one. Finite probability vectors

[F5]

The nonnegative integral is monotone and homogeneous, and is additive on nonnegative measurable functions. Monotonicity and nonnegative homogeneity of the nonnegative integral Additivity of the nonnegative Lebesgue integral

[F6]

A metric is symmetric and satisfies the triangle inequality. Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric

[F7]

The metric cost is continuous on the product: the triangle inequality gives $∣ d (x, y) - d (x^{'}, y^{'}) ∣ \leq d (x, x^{'}) + d (y, y^{'})$ , and a continuous real-valued function is Borel. Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space The Borel sigma-algebra of a topological space

[F8]

A nonnegative function constant on each member of a finite measurable partition is simple, and its nonnegative integral is the finite sum of its values times the corresponding cell measures. Nonnegative simple measurable functions The integral of a nonnegative simple function The nonnegative integral agrees with the simple integral on simple functions

Proof

technique · direct

1.1F3F4givenalgebra

If $p_{i} = 0$ , [F3] and nonnegativity imply $π_{i j} = 0$ for every $j$ ; similarly, if $q_{j} = 0$ , then $π_{i j} = 0$ for every $i$ . For $p_{i} > 0$ define $μ_{i} (A) = μ (A \cap C_{i}) / p_{i}$ for Borel $A \subseteq X$ , and for $q_{j} > 0$ define $ν_{j} (A) = ν (A \cap C_{j}) / q_{j}$ . Countable additivity of $μ, ν$ is preserved by restriction to $C_{i}, C_{j}$ and multiplication by $1 / p_{i}, 1 / q_{j}$ ; each restricted measure has total mass one after normalization. Hence these are Borel probability measures concentrated on $C_{i}, C_{j}$ .

2.1F1F2F3F4step 1.1construct

Each conditional probability measure in step 1.1 is sigma-finite because $X$ has mass one. If $X$ is compact, the Borel section-product construction in [F2] applied to the probabilities $μ_{i}, ν_{j}$ on the full compact space $X$ gives a Borel product probability for each active pair; the cells themselves need not be compact. If $X$ is second-countable, [F1] and the ordinary product construction in [F2] give the same conclusion. Denote the chosen construction in either case by $μ_{i} \times ν_{j}$ . Define $γ : = \sum_{i, j : p_{i}, q_{j} > 0} π_{i j} (μ_{i} \times ν_{j})$ . Since zero rows and columns contribute zero by step 1.1, its total mass is $\sum_{i, j} π_{i j} = \sum_{i} p_{i} = 1$ . Thus $γ$ is a Borel probability measure.

3.1F2F3F4step 2.1algebra

For Borel $A \subseteq X$ , the rectangle formula [F2] and row sums give $γ (A \times X) = \sum_{i : p_{i} > 0} \sum_{j : q_{j} > 0} π_{i j} μ_{i} (A) = \sum_{i : p_{i} > 0} p_{i} μ_{i} (A) = \sum_{i} μ (A \cap C_{i}) = μ (A)$ . The same calculation using column sums gives $γ (X \times A) = ν (A)$ , so $γ$ has the required marginals.

4.1step 1.1step 2.1F2F5F6F7F8algebra∎

The cost $d (x, y)$ is nonnegative Borel by [F7]. For each active pair with $p_{i}, q_{j} > 0$ , [F2] gives $(μ_{i} \times ν_{j}) (C_{i} \times C_{j}) = 1$ . For every other active pair $(k, ℓ) \neq (i, j)$ , disjointness of the partition forces $μ_{k} (C_{i}) = 0$ or $ν_{ℓ} (C_{j}) = 0$ , so [F2] gives its product measure zero on $C_{i} \times C_{j}$ . Consequently the finite measure sum of step 2.1 gives $γ (C_{i} \times C_{j}) = π_{i j}$ . On zero rows or columns both sides are zero by step 1.1. On each active block, [F6] gives $d (x, y) \leq d (x, x_{i}) + d (x_{i}, x_{j}) + d (x_{j}, y) < d (x_{i}, x_{j}) + 2 ε$ . The finite Borel step function $s (x, y) : = \sum_{i, j} (d (x_{i}, x_{j}) + 2 ε) 1_{C_{i} \times C_{j}} (x, y)$ majorizes $d (x, y)$ on all blocks, including zero-mass ones. Monotonicity [F5], the block masses just computed and the simple-integral formula [F8] give $\int d d γ \leq \int s d γ = \sum_{i, j} (d (x_{i}, x_{j}) + 2 ε) π_{i j} = \sum_{i, j} π_{i j} d (x_{i}, x_{j}) + 2 ε$ .

Diagram

For a Borel quantization map $Q : X \to F$ , both paths in the square send $(x, y)$ to $Q (x)$ ; the second-coordinate square is analogous. The diagram records coordinate compatibility only; the proof of the coupling and cost estimates above is independent of it.

diagram failed to render:
X\times X \arrow[r,"Q\times Q"] \arrow[d,"\mathrm{pr}_1"'] & F\times F \arrow[d,"\mathrm{pr}_1"] \\
X \arrow[r,"Q"'] & F

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Used by

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Sources

Dmitriy Panchenko, Lecture Notes on Probability Theory, §4.4 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.1, PDF pp. 3–4 (standard reference, not scraped)