Existence of a positive smooth volume density of total mass one on a smooth Lorentzian four-manifold

Given: A nonempty time-oriented smooth Lorentzian four-manifold $(M,g,N)$ and ${\mathrm{A}\mathrm{C}}_{\omega }$.

$M$ is a smooth four-manifold: Hausdorff, second countable, locally Euclidean, and coordinate balls form a basis; consequently there is a countable open cover by relatively compact coordinate charts, and for every such cover there is a smooth partition of unity subordinate to it, locally finite and with values in $[0,1]$ summing to one (Smooth manifolds and their smooth charts, Coordinate balls form a basis of a topological manifold, Smooth partitions of unity subordinate to an open cover, Smooth partitions subordinate to a countable coordinate cover, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Densities as coefficients: a nonnegative density on $M$ assigns to every chart a Borel coefficient taking values in $[0,\mathrm{\infty }]$ and obeying the absolute-Jacobian overlap law; products of coefficients with nonnegative smooth functions and locally finite sums obey the same law, so the product of a nonnegative smooth function with a density and every locally finite sum of densities are again densities; a density is smooth when all its chart coefficients are smooth (Pointwise Borel nonnegative densities). In arbitrary overlapping charts with transition $x=\phi \circ y$, tensor transformation gives $g_y=(D\phi {)}^T(g_x\circ \phi )D\phi$, hence $\sqrt{\mathrm{\mid }\det g_y\mathrm{\mid }}=\mathrm{\mid }\det D\phi \mathrm{\mid }\sqrt{\mathrm{\mid }\det (g_x\circ \phi )\mathrm{\mid }}$. Thus the coefficients $w=\sqrt{\mathrm{\mid }\det g_{ab}\mathrm{\mid }}$ obey the density overlap law and define a positive smooth density without requiring an orientation. This is a determinant identity; the oriented volume-form assertion is not used (The metric volume form, current flux through 3-faces and finite-chain charges clause (a) for the coefficient formula).

Gluing and measure: a nonnegative density together with a countable locally finite chart cover and a subordinate smooth partition of unity defines a countably additive nonnegative Borel measure ${\mu }_r$ by the displayed chart formula, independent of the cover and partition; for a positive smooth density ${\mu }_r$ is finite on compact sets, locally finite and $\sigma$-finite, and regular Borel (Countable partition construction of the Borel set function, The glued set function is a Borel measure, Intrinsic density measure and its chart restriction, Positive smooth densities give Radon volume).

Positivity: for a positive smooth density ${\mu }_r(O)>0$ for every nonempty open $O$, and $r>0$ on a nonempty open set means the coefficient in any chart meeting that set is positive on a nonempty open subset of the chart image (Positive open-set and metric-ball volume, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

Chart formula: for a nonnegative Borel $f$ and any chart partition $(x_i,{\phi }_i)$ one has ${\int }_{M}fd{\mu }_r={\sum }_i{\int }_{x_i(U_i)}({\phi }_{i}fr{)}_{x_i}d{\lambda }_4$, with $0\cdot \mathrm{\infty }=0$; in particular the measure is additive over finite chart covers and the map $r\mapsto {\mu }_r$ is positively homogeneous: ${\mu }_{cr}=c{\mu }_r$ for $c>0$ (Measurable integration extends smooth density integration).

Finiteness and convergence: a bounded Borel function supported in a coordinate box has finite Lebesgue integral, its integral over a Lebesgue-null Borel set vanishes, and the integral of a locally finite sum of nonnegative integrands is the sum of the integrals; a nonnegative measurable function with vanishing integral vanishes almost everywhere, and a $C^1$ diffeomorphism between open subsets of ${\mathbb{R}}^4$ carries Lebesgue-null sets to Lebesgue-null sets (Borel Darboux integrands in finite dimension, Monotonicity and nonnegative homogeneity of the nonnegative integral, A nonnegative integral over a null set vanishes, Monotone convergence for the integral, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, A C^1 diffeomorphism maps Lebesgue null sets to Lebesgue null sets).