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Finite order statistics and rank maps are measurable

Statement

Let $(Ω, F)$ be a measurable space, let $R_{1}, \dots, R_{m}$ be measurable real-valued functions with $m \geq 1$ , and let $1 \leq k \leq m$ . The $k$ th order statistic $R_{(k)}$ is measurable. For a finite real score vector $R_{1}, \dots, R_{n}$ and another measurable real-valued function $U$ on the same space, its upper-rank p-value and tie-randomized rank are measurable.

Facts & Assumptions

Given: The measurable real score functions, integer bounds on $k$ , and, for the randomized formula, a measurable $U : [0, 1]$ -valued input on the same measurable space.

[F1]

Finite maxima and minima, finite sums and scalar multiples of measurable real functions are measurable. Arithmetic and lattice operations preserve measurability whenever they are defined

[F2]

Real measurability is preimage measurability for the Borel sigma-algebra. A measurable function between measurable spaces

[F3]

The Borel sigma-algebra contains closed intervals and half-lines because it is generated by open sets. The Borel sigma-algebra of a topological space

[F4]

The conformal p-value and tie-randomized p-value are the finite rank formulas in the definition. Conformal scores and rank p-values

Proof

1.1F1given

For each $k$ -element subset $I \subseteq {1, \dots, m}$ , let $M_{I} = \max_{i \in I} R_{i}$ . Then $R_{(k)} = \min_{∣ I ∣ = k} M_{I}$ : every $k$ -element subset has maximum at least the $k$ th smallest score, and the subset of the $k$ smallest scores has that maximum exactly. There are finitely many such nonempty subsets, so [F1] makes this finite minimum of finite maxima measurable.

1.2F1F2F3F4

For any two indices, ${R_{i} \geq R_{j}} = {R_{i} - R_{j} \in [0, + \infty)}$ is measurable by [F1]–[F3]. The upper-rank p-value is a finite sum of indicators of these events divided by $n$ , so is measurable.

2.1F1F4step 1.2∎

On the same domain, the tie count is a finite sum of the comparison indicators from step 1.2. Multiplication by the measurable real function $U$ , addition, and division by the positive integer $n$ preserve measurability by [F1]. Hence the randomized rank map is measurable.

Depends on

Used by

Full conformal prediction set Algorithm
Split conformal prediction set Algorithm
Conformal rank superuniformity under exchangeability Theorem

Dependency tree · two levels

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Sources

John K. Hunter, Measure Theory, §§3.1 and 5.1 (standard reference, not scraped)