Existence of regular Lypanuov function for any probability measure Given a Borel probability measure $\mu$ on $\mathbb{R}^d$, there exists (since any such $\mu$ is tight) a function $f$ with compact sublevel sets such that $\int f d\mu < \infty$. Such a function is, for example, constructed as a piecewise constant step function, where the height and the location and size of each step is determined by the (sufficiently fast shrinking) size of the measure on growing balls (this construction works due to tightness). My question is the following: Does there always exist a $C^2$-function $f$ (of course depending on $\mu$) with the above integrability condition, which has bounded first and second derivative?
Thanks in advance!
 A: The answer seems to be affirmative:
Indeed, the precise construction of a piecewise constant, nonnegative Lyapunov function for given $\mu$ is presented neatly in the answer to this post. Clearly, the construction works in the multidimensional case as well (note that the above post is concerned with probability measures on $\mathbb{R}$ only); the constructed function is radial. Without loss of generality, assume (using notation from the linked post) $R_{j+1} > R_j +1$. Then, define $g =1$ on $B_{R_2}$. On $B_{R_3}\backslash B_{R_2}$, set $g(x) = h(|x|)$, with $h \in C^2([R_2,R_3])$ increasing such that $h(R_2) =1, h_2(R_3)=2$ with all partial derivatives of $h_2$ bounded. Clearly, $g \leq f$ (f being the function constructed in the above link) on $B_{R_3}$. Iterating this constrution yields a non-negative, increasing (radial) $C^2$-function $g: \mathbb{R}^d \to \mathbb{R}$ with compact sublevel sets with $g \leq f$ pointwise. Since by assumption each intervall $[R_j,R_{j+1}]$ has length at least $1$, the functions $h_j$ used to define $g$ (the iterative analogues to $h_2$ above) may be chosen to have uniformly (in $j$) bounded first and second partial derivatives. Thus, we are done.
