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Gaussian integrability of distance function under the Lyapunov condition

In this note we give a direct proof of the Gaussian integrability of distance function as $μe^{δd^2(x,x_0)} < \infty$ for some $δ>0$ 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].