I am trying to answer part (a) problem from Axler's book, Measure, Integration and Real Analysis (seen below). I think I'm almost there but I'm stuck on a certain case.
We also have the following equivalences from a theorem in the book:
Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$. If we let Inn denote the inner measure function, it's easy to see from the definitions that Inn$(A)\leq|A|$.
If $A$ is bounded and $|A|<\infty$,
then $\exists$ a closed, bounded set $F$ s.t. $F\subset A$ and $|F|<\infty$ and $|A\setminus F|<\epsilon$ (using equivalence (b) from above).
$\implies |A|<|F|+\epsilon$ (using $|A\setminus F|=|A|-|F|$).
$\implies |A|\leq$ Inn$(F)+\epsilon$.
$\implies |A|\leq$ Inn$(F)$.
The issue now is what to do if ${\mathbf A}$ is unbounded or ${\mathbf |A|=\infty}$ (possibly both at the same time). We can't use $|A\setminus F|=|A|-|F|$ like before or say that $F$ is bounded like before.
Below I include Axler's definition of outer measure, in case it is not standard:
