  In this note we give a direct proof of the Gaussian integrability of distance
function as $\mu e^{\delta d^2(x,x_0)} < \infty$ for some $\delta>0$ provided
the Lyapunov condition holds for symmetric diffusion Markov operators, which
answers a question proposed in Cattiaux-Guillin-Wu [6, Page 295]. The similar
argument still works for diffusions processes with unbounded diffusion
coefficients and for jump processes such as birth-death chains. An analogous
discussion is also made under the Gozlan's condition arising from [9,
Proposition 3.5].
