  We construct the entropic measure $\mathbb{P}^\beta$ on compact manifolds of
any dimension. It is defined as the push forward of the Dirichlet process
(another random probability measure, well-known to exist on spaces of any
dimension) under the {\em conjugation map}
$$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map is a continuous
involution. It can be regarded as the canonical extension to higher dimensional
spaces of a map between probability measures on 1-dimensional spaces
characterized by the fact that the distribution functions of $\mu$ and
$\Conj(\mu)$ are inverse to each other. We also present an heuristic
interpretation of the entropic measure as
$$d\mathbb{P}^\beta(\mu)=\frac{1}{Z}\exp(-\beta\cdot {Ent} (\mu|m))\cdot
d\mathbb{P}^0(\mu).$$
