  The property of measure concentration is that an arbitrary 1-Lipschitz
function $f:X\to \mathbb{R}$ on an mm-space $X$ is almost close to a constant
function. In this paper, we prove that if such a concentration phenomenon
arise, then any 1-Lipschitz map $f$ from $X$ to a space $Y$ with a doubling
measure also concentrates to a constant map. As a corollary, we get any
1-Lipschitz map to a Riemannian manifold with a lower Ricci curvature bounds
also concentrates to a constant map.

Metric Geometry