Consider the measure $\mu $ on $R$ such that $\mu ([-r,r])>0$ for all $r>0$.

Can we construct a (smooth) function $f$ satisfying $\mu -fdx\ge 0$ in a measure sense? If $\mu $ has a continuous density $g$, then it seems easy. But what conditions are needed for the existence of $f$ for a measure $\mu $? or is it possible always?

Can we construct a (smooth) function $f$ satisfying $\mu -fdx\ge 0$ in a measure sense? If $\mu $ has a continuous density $g$, then it seems easy. But what conditions are needed for the existence of $f$ for a measure $\mu $? or is it possible always?