Fix $(X, g)$ to be some compact Riemannian manifold and $V \in \Gamma(TX)$ a smooth, non-vanishing vector field. Suppose the flow is *minimal*, i.e. every orbit is dense in $X$, and *volume-preserving*.

Suppose there exists a measure $\sigma$ that is both invariant under the flow of $V$ and absolutely continuous with respect to the Riemannian volume measure. In other words, it is given by integration against $f \text{dvol}_g$ for $f \in L^1(X)$.

If $f$ is not merely $L^1$ but continuous, then our topological assumptions imply immediately that $f$ is constant, so actually any invariant measures are a constant multiple of the volume measure (the minimality precludes the existence of any singular invariant measures).

Can we construct an example such that $f$ is *not* constant almost everywhere?