CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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The zero-measure Lp boundary

Statement refuted

For every finite compactly supported measure $μ$ and every $1 < p < \infty$ , a single-hidden-layer network family is dense in $L^{p} (μ)$ only if its activation is not polynomial almost everywhere.

Statement references

Leshno et al., Theorem 1 and $L^{p}$ discussion

Facts & Assumptions

Given: The zero measure $μ = 0$ , an exponent $1 < p < \infty$ , and the constant activation $σ (t) = 1$ .

[F1]

$L^{p} (μ)$ is the space of finite- $p$ -integral functions modulo equality almost everywhere (The function space $L^{p} (μ)$ for $0 < p < \infty$ , The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

The single-hidden-layer family includes the zero function by taking width zero and output bias zero (Single-hidden-layer network realization).

[F3]

A measure is a nonnegative countably additive set function on its sigma-algebra; the zero measure assigns zero to every measurable set (Measures on sigma-algebras).

Counterexample

1.1givenF3algebra

The constant activation $σ (t) = 1$ is a polynomial, and the zero measure is finite, compactly supported, and absolutely continuous with respect to Lebesgue measure by [F3].

1.2givenF1

Since every measurable function has $L^{p} (0)$ seminorm zero, the quotient space $L^{p} (0)$ contains only its zero class by [F1].

2.1step 1.1step 1.2F2∎

The network family contains the zero function by [F2], whose class is the only element of $L^{p} (0)$ . Thus it is dense despite the polynomial activation, refuting the unrestricted “only if” claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, Theorem 1 and Lp discussion (standard reference, not scraped)