A set $\Gamma$ of probability measures on $X$ is said to be tight if for every $\epsilon > 0$ there exists a compact set $K_\epsilon$ of $X$ such that
$\mu(K) \geq 1 - \epsilon \text{ for all $\mu \in \Gamma$}.$
Fine, now let $\mathcal M$ be the set of measurable mappings on the probability space $(\Omega, \mathcal F, P)$ if there exists $M > 0$ such that
$\int_\Omega |f(\omega)| \, dP(\omega) \leq M \text{ for all $f \in \mathcal M$}.$
How do I now show that the set $\{f_{\#}P : f \in \mathcal M\}$ is tight?
Please only give me a hint as this is "homework". What I tried to do was assume it is not tight and then try to construct a compact set that gives me a contradiction, but that seems to lead nowhere.