How to proof that $\mathcal{P}(X)_{\tau}=\mathcal{P}(X)$ on the measure space $(X,\mathcal{P}(X))$ when $\tau$ is the counting measure.
My teacher has defined: $$\mathbb{N}_{\tau}=\{N\subseteq X\; |\; \text{there exist }P\in\mathcal{P}(X)\;\text{so}\;N\subseteq P\;\;\text{and}\;\tau(P)=0\},$$ where I see that $\mathbb{N}_{\tau}=\{\emptyset\}$, but I don't know how to understand in terms of the Powerset.
I only seem to understand this intuitively, as if $\mathcal{P}(X)$ contain all subsets of $X$, and $\tau$ can be defined on all of the subsets, they must be equal to each other.
I have hard time understanding what my professor would like us to use in this scenario - I thought about defining $\mathcal{P}(X)$ as point masses, however, I have to show that they contain the same sets.
Any help proofing this, would be greatly appreciated.