MathOverflow

I d be interested in books or papers that treat universally measurable functions and in particular integration theory of universally measurable functions. As a bonus it would be nice if it would also treat regular conditional probabilities/Markov kernels...

It is known that, if $\rho$ is a $\alpha-$strongly log-concave density on $R^d$, then $P_t \rho = \rho * N(0,t I)$ is $\frac{\alpha}{1+t \alpha}-$strongly log-concave, for $P_t$ the heat semigroup.
I wonder if such a property holds for the log-smoothness of the measure - that is, if $ \nabla \log \rho$ is $L-$Lipschitz, then what can we say about $\nabla \log (P_t \rho)$? Is it also Lipschitz, and if so what is the constant?...

A Hessian manifold $(M,D,g)$ is a manifold with a flat symmetric connection $D$ and a Riemannian metric $g$ that is locally expressed, in flat charts of $D$, as the Hessian with respect to $D$ of some function (called a local potential).
For no reason I was trying to see if there is a simply connected Hessian manifold whose metric is complete and of constant negative curvature, i.e. isometric to the hyperbolic plane. Reversing the viewpoint, this is equivalent to the question: can the...

We define the angular defect function for a triangle with angles $A,B,C$ ($A+B+C=\pi$):
$$\Phi(A,B,C) = \frac{1 + \cos A \cos B \cos C}{\sqrt{3}\,\sin A \sin B \sin C}.$$
For a point $P$ inside triangle $\triangle ABC$, let $\mathcal{E}(P) = PA+PB+PC - 2(d_a+d_b+d_c)$ be the Erdős–Mordell gap....