Energy-based density and partition function

Let $(X,\mathcal{A},\mu )$ be a measure space, let $\mathrm{\Theta }$ be a parameter set, and for each $\theta \in \mathrm{\Theta }$ let $E_{\theta }:X\to \mathbb{R}$ be a measurable energy function. Its partition function relative to $\mu$ is $Z_{\theta }:={\int }_X{e}^{-E_{\theta }(x)}d\mu (x)\mathrm{.}$ The admissible parameter set is ${\mathrm{\Theta }}_{\mu }:=\{\theta \in \mathrm{\Theta }:0<Z_{\theta }<\mathrm{\infty }\}\mathrm{.}$ For $\theta \in {\mathrm{\Theta }}_{\mu }$, the energy-based probability measure is the measure with density $e^{-E_{\theta }}\mathrm{/}{Z}_{\theta }$ relative to $\mu$: $P_{\theta }(A):=\frac{1}{Z_{\theta }}{\int }_A{e}^{-E_{\theta }(x)}d\mu (x),A\in \mathcal{A}\mathrm{.}$ The restriction $0<Z_{\theta }<\mathrm{\infty }$ is part of the definition; parameters outside ${\mathrm{\Theta }}_{\mu }$ do not define a normalized model by this formula.

Measure spaces defines the triple $(X,\mathcal{A},\mu )$ used here.

A measurable function between measurable spaces defines measurability of $E_{\theta }$.

The nonnegative Lebesgue integral defines the integral used for $Z_{\theta }$.

The measure with density $f$ relative to $\mu$ defines the measure with density $e^{-E_{\theta }}$ relative to $\mu$.

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

An admissibly normalized energy density is a probability measure ↗ establishes the normalization's well-definedness.