Let $\mathscr{A}$ be a family of probability measures then this family is tight iff there exists a function $f\in C(\mathbb{R^n})$ such that $f(x)\to\infty$ as $|x|\to\infty$ and $$\sup_{\mu\in\mathscr{A}}\int f\,\mathrm{d}\mu<\infty.$$
I think the function needs to be nonnegative. If that is the case I can prove the tightness of the family but again I am unable to show the existence of such a function from the tightness. Any help will be appreciated.
Best Answer
If the family is tight there is a sequence $n_k$ increasing to $\infty$ such that $\mu \{x:n_n \leq\|x\| <n_{k+1}\} <\frac 1 {2^{k}}$ for all $\mu$ in the family. . Let $f(x)=g(\|x\|)$ where $g$ is a piece wise linear function on $[0,\infty)$, linear on each of the intervals $[n_k,n_{k+1})$ taking the value $k$ at $n_k$. Note that $g(x) \leq k+1$ on $[n_k, n_{k+1})$. It follows that $\int f d \mu \leq \sum_k \frac {k+1} {2^{k}}$ for all $\mu$.