Outer regular measure

Remarks

Every open set is outer regular, by taking $U = A$.

Example: Lebesgue measure $\lambda$ on the Borel sets of $\R$ is outer regular. For a Borel set $A$ and $\varepsilon > 0$, the outer measure construction gives intervals $I_k$ covering $A$ with $\sum_k \lambda(I_k) < \lambda(A) + \varepsilon$. Enlarging each $I_k$ slightly to an open interval gives an open $U \supseteq A$ with $\lambda(U) < \lambda(A) + 2\varepsilon$.

Non-example: counting measure $\mu$ on $\R$ is not outer regular. The set $\{0\}$ has $\mu(\{0\}) = 1$, but every open set containing $0$ is infinite and has measure $\infty$.