A Dirac measure is a measure $\delta_x$ on a set $X$ defined for a given $x\in X$ and any set $A\subseteq X$ by $$\delta_x(A)=\begin{cases}0, & x\notin A, \\ 1, &x\in A.\end{cases}$$
I wonder whether it is possible to construct some measure $\tilde{\delta}_x$ such that $\tilde{\delta}_x(A)=1$ if $x\subseteq A$ and $\tilde{\delta}_x(A)=0$ if $x\not\subseteq A$, where $x\subseteq X$ is a compact set.