LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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An admissibly normalized energy density is a probability measure

Statement

For every measure space $(X, A, μ)$ , measurable real energy $E_{θ}$ , and parameter $θ$ with $0 < Z_{θ} : = \int_{X} e^{- E_{θ}} d μ < \infty$ , the normalized measure $P_{θ} (A) : = Z_{θ}^{- 1} \int_{A} e^{- E_{θ} (x)} d μ (x), A \in A,$ is a probability measure on $(X, A)$ .

Statement references

Energy-based density and partition function supplies the notation $Z_{θ}$ and $P_{θ}$ .

The measure with density $f$ relative to $μ$ identifies the nonnegative density integral as a measure.

Probability measures and probability spaces defines a probability measure by total mass one.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ , a measurable real energy $E_{θ}$ , and $0 < Z_{θ} < \infty$ .

[F1]

Energy-based density and partition function: $P_{θ} (A) = Z_{θ}^{- 1} \int_{A} e^{- E_{θ}} d μ$ and admissibility requires $0 < Z_{θ} < \infty$ .

[F2]

The measure with density $f$ relative to $μ$ : if $f \geq 0$ is measurable, then $A \mapsto \int_{A} f d μ$ is a measure.

[F3]

Probability measures and probability spaces: a measure on $(X, A)$ with total mass one is a probability measure.

[F4]

The real exponential function and the number $e$ by a power series and The exponential series converges absolutely for every real argument: the real exponential is the power-series sum $\exp (t) = \sum_{n \geq 0} t^{n} / n!$ , with infinite radius of convergence.

[F5]

The sum of a real power series is continuous at every point strictly inside its interval of convergence: a power-series sum is continuous at every point strictly inside its radius of convergence.

[F6]

A continuous map has Borel preimages of Borel sets: a continuous real map has Borel preimages of Borel sets.

[F7]

Composition with a Borel measurable outer map preserves measurability: a measurable map composed with a Borel measurable outer map is measurable.

[F8]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (t) > 0$ for every real $t$ .

Proof

1.1givenF1F2F4F5F6F7F8algebra

Define $ϕ (t) : = \exp (- t)$ . By [F4], $ϕ (t) = \sum_{n \geq 0} (- 1)^{n} t^{n} / n!$ is a power-series sum with infinite radius, so [F5] makes $ϕ$ continuous on $R$ . Thus [F6] makes it Borel measurable, and [F7] with the measurable energy from [F1] makes $x \mapsto \exp (- E_{θ} (x))$ measurable. Its values are positive by [F8], hence nonnegative. Put $M_{θ} (A) : = \int_{A} e^{- E_{θ}} d μ$ . Now [F2] makes $M_{θ}$ a measure and $M_{θ} (X) = Z_{θ}$ . Thus $P_{θ} (\emptyset) = Z_{θ}^{- 1} M_{θ} (\emptyset) = 0$ . For any pairwise disjoint sequence $(A_{n})_{n \geq 0}$ , [F2] and finite-sum homogeneity give $P_{θ} (⋃_{n \geq 0} A_{n}) = Z_{θ}^{- 1} \sum_{n \geq 0} M_{θ} (A_{n}) = \lim_{N \to \infty} \sum_{n = 0}^{N} Z_{θ}^{- 1} M_{θ} (A_{n}) = \sum_{n \geq 0} P_{θ} (A_{n})$ . Therefore $P_{θ}$ is a measure.

2.1F1F3step 1.1algebra∎

Its total mass is $P_{θ} (X) = Z_{θ}^{- 1} M_{θ} (X) = 1$ , so [F3] makes $P_{θ}$ a probability measure.

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Energy-based density and partition function.

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Sources

Aapo Hyvärinen, Estimation of Non-Normalized Statistical Models by Score Matching, Sections 1–2 (standard reference, not scraped)
Kevin P. Murphy, Probabilistic Machine Learning: Advanced Topics, Chapter 24 (standard reference, not scraped)