Does a locally $\mu$-null set have measure 0?

A set $N$ is called locally $\mu$-null if for each set $A$ that belongs to $\mathscr{A}$ and satisfies $\mu(A)<+\infty$ the set $A\bigcap N$ is $\mu$-null. I have always been in trouble understanding this concept intuitively. I would like to ask is there an example of a locally $\mu$-null set to have positive measure?

Thanks a lot for any help!