Two short proofs of the measure rigidity theorem

In this talk I will show how the general existence of transport maps and the properties of interpolation measures can be used to show that any two reference measures with the measure contraction property must be mutually absolutely continuous.

The first proof uses the fact that non-degeneracy of the reference measure implies that geodesics issuing from a fixed point $x\in M$ and ending in $y\in M$ can be extended to a minimal geodesic beyond $y$ for almost all $y\in M$. The second proof uses a characterization of the interpolation densities and applies also to a more general setting.

The result implies, in particular, that any two measures $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ on a metric spaces $(M,d)$ must be mutually absolutely continuous whenever $(M,d,\mathrm{m}_{1})$ and $(M,d,\mathrm{m}_{2})$ are $\mathrm{RCD}^{*}(K,N)$-spaces with $K\in\mathbb{R}$ and $N\in[1,\infty)$.