Vacuum-population, measure and selection data

Fix a measurable vacuum set $V$ drawn from an explicitly specified ensemble, with sigma-algebra $\mathrm{\Sigma }$ on $V$. A vacuum population measure is a positive countably additive measure $\mu :\mathrm{\Sigma }\to [0,1]$ with $\mu (V)=1$; it describes the prescribed population process or prior and is not the uniform count over vacua unless that equality is separately adopted. Let $(\mathcal{D},\mathcal{A},\nu )$ be a data space with a declared sigma-finite reference measure $\nu$. For fixed observed data $D\in \mathcal{D}$, supply as part of the model a measurable function ${\mathrm{\ell }}_D:V\to [0,\mathrm{\infty })$, called the likelihood weight; this definition does not derive it from a family of data laws. If ${\mathrm{\ell }}_D(v)$ is claimed to be a density, the input must additionally specify a family of probability laws $(Q_v{)}_{v\in V}$ on $(\mathcal{D},\mathcal{A})$ with $Q_v\ll \nu$ and a selected pointwise measurable version of $d{Q}_v\mathrm{/}d\nu$ at $D$. Such a density is determined only up to $\nu$-null sets for each $v$, so a fixed-$D$ evaluation uses the supplied representative; its units are inverse $\nu$-units. A selection function $S:V\to [0,1]$ is measurable and records the probability or weight assigned by the declared observer/selection model. Write

When $0<Z(D,S)<\mathrm{\infty }$, define the selected probability measure by

The input data are a specified vacuum set, population dynamics or prior, likelihood, data, selection rule, and any regulator and its removal procedure. The $P$ above is a normalized conditional model prediction; it is not implied by a finite vacuum count or a flux-vacuum density.

Flux-vacuum ensemble and tadpole-bound data