LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The compact compander is a measurable finite quantizer

Statement

For the data in Positive C1 compact one-dimensional compander, every cell is Borel, the cells partition $[a, b]$ , the index map is measurable, and $D_{N} (f, g)$ is finite for each $N \geq 1$ .

Facts & Assumptions

Given: $a < b$ , positive normalized $C^{1}$ densities $f, g$ on $[a, b]$ , and an integer $N \geq 1$ .

[F1]

Positive C1 compact one-dimensional compander: $β_{j} = ϕ (j / N)$ are the cell boundaries, $r_{j} = ϕ ((j - \frac{1}{2}) / N)$ are the reproduction points, and the cells are half-open except for the final cell containing $b$ .

[F2]

A positive C1 point density has a C2 cumulative inverse: $ϕ$ is a strictly increasing bijection from $[0, 1]$ to $[a, b]$ .

[F3]

Finite measurable squared-error quantizer and nearest-center rule: a finite quantizer is a Borel map into a finite discrete index set and its distortion is the integral of squared reconstruction error.

Proof

1.1givenF1F2

The boundary sequence $β_{0} < \dots < β_{N}$ is strictly increasing by [F2], with $β_{0} = a$ and $β_{N} = b$ . Thus the intervals in [F1] are pairwise disjoint Borel sets whose union is $[a, b]$ .

2.1givenF1F2F3step 1.1algebra∎

Each index preimage is one of those Borel intervals, so the finite-valued index map is Borel and defines a quantizer as in [F3]. Each reproduction point lies in $[a, b]$ by [F1] and [F2], so $∣ x - r_{j} ∣ \leq b - a$ on cell $j$ ; hence $0 \leq D_{N} (f, g) \leq (b - a)^{2} \int_{a}^{b} f (x) d x = (b - a)^{2} < \infty$ .

Depends on

Used by

Exact high-rate limit for a positive compact one-dimensional compander Theorem

Cited to discharge well-definedness by Positive C1 compact one-dimensional compander.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)