Is there an outer measure $\mu$ on $\mathbb{R}$ where $\mu(\mathbb{R})=1$ and all subsets of $\mathbb{R}$ are $\mu$ measurable

Question

I'm looking for an outer measure $\mu$ on $\mathbb{R}$ such that $\mu(\mathbb{R})=1$ and where the set of measurable(Caratheodory) subsets is $\cal{P}(\mathbb{R})$. My first instinct was to use a measure that assigns every nonempty set 1, but then the only measurable subsets are $\mathbb{R}, \emptyset$. I've also tried a measure that assigns countable sets 0 and uncountable sets 1 but uncountable sets aren't measurable like this.