 Solution

Fix an orientation and a positive volume form $\Omega$. Define the divergence by the Lie derivativeA smooth function is a flow coboundary when it equals $Fu$ for some smooth $u$. Every other positive volume form is $e^h\Omega$ for a smooth $h$. The product rule for the Lie derivative givesThus changing the reference volume form changes its divergence by a flow coboundary. If volume forms of either orientation are allowed, take the logarithm of the absolute ratio on each connected component; multiplying a reference form by a constant sign leaves its divergence unchanged.

If ${\mathrm{div}}_{\Omega }F=Fu$, the preceding product rule showsConversely, if $e^h\Omega$ is invariant, then ${\mathrm{div}}_{\Omega }F=-Fh=F(-h)$ is a flow coboundary. Vanishing of the Lie derivative is equivalent to flow invariance, since ${\partial }_t{\varphi }_t^{\ast }\Omega ={\varphi }_t^{\ast }({\mathcal{L}}_F\Omega )$. HenceThe equivalence holds for every choice of reference volume form, proving the invariant volume criterion for a smooth flow. The smoothness requirement matters: a merely measurable transfer function does not give a smooth invariant volume form.