Definition. Integrable observables and Gaussian expectation [fdip-0006]
Definition. Integrable observables and Gaussian expectation [fdip-0006]
For the Gaussian probability law \(\mu _Q\) on \((E,h)\), an integrable observable is a Borel function \(F:E\to \mathbb R\) with \(\int _E|F|\,d\mu _Q<\infty \). Its Gaussian expectation is \[\begin {aligned} \omega _Q(F)=\langle F\rangle _Q&=\int _E F\,d\mu _Q\\ &=Z_Q^{-1}\int _E F(x)w_Q(x)\,d\lambda _E(x). \end {aligned}\] The equality follows first for nonnegative simple functions from the density definition, then by monotone convergence, and finally by positive and negative parts for an absolutely integrable function.
These actual functions form a real vector space: the pointwise bound \(|aF+bG|\leq |a||F|+|b||G|\) proves closure, and integration is real linear on it. One may instead take almost-everywhere classes when using \(L^1\); polynomial observables here remain actual functions. All polynomial moments give its polynomial subalgebra; the state proposition proves normalization, positivity and Hermiticity there. The integrable-domain example explains why the full vector space need not be an algebra.
This extends finite expectation: absolute integrability replaces automatic convergence of a finite sum. The real-spin convention is [friedli2017statistical, Definition 8.1 and Equation (8.3), p. 381]; the density-integral extension uses [tao2011measure, Theorem 1.4.44, pp. 107-108, preliminary version]. Constants and oddness give the vanishing linear means. Covariance and variance extend the finite convention of Notation [ftip-008H]; the partition function, two-point expectation and other observable expectations remain distinct quantities.