Plain-English statement
The unnormalized Gibbs weight associated with a potential V is exp(-V), represented as an extended nonnegative real density.
Mathematical statement
g_V(x) = exp(-V(x)), encoded as ENNReal.ofReal(exp(-V(x))).
Intuition
withDensity and lintegral use ENNReal-valued densities. The wrapper keeps positivity and possible infinite normalization in the measure-theoretic type rather than prematurely coercing to real numbers.
Conditions
- No measurability or integrability is built into the definition.
- A measurable-space structure is introduced only by later lemmas.
Why these conditions cannot be dropped
- A formal Gibbs expression is not automatically measurable.
- The normalization integral may be zero or infinite until separate theorems rule those cases out.
Proof route
- Define the ENNReal density pointwise.
- Prove positivity, finiteness, and measurability as separate leaves.
- Use envelope or coercivity lemmas to prove finite nonzero normalization.
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Lean statement
noncomputable def gibbsDensityENNReal (V : α → ℝ) : α → ℝ≥0∞ :=
fun x => ENNReal.ofReal (Real.exp (-V x))
/-- Gibbs densities are pointwise positive. -/Lean interface notes
- ENNReal.ofReal is used because Measure.withDensity expects an ENNReal density.
- Real.exp is nonnegative, so the coercion preserves the intended value.
- Definition-level compilation does not certify normalizability.