I‘m reading an Analysis script, where the following is stated and the proof is left as a practice for the reader.
Let $M\subseteq \mathbb{R} ^{n}$ be a Lebesgue measurable set. Then $\lambda \left( M\right) =\inf \left\{ \lambda \left( U\right) |U\supseteq M, U open\right\} =\sup \left\{ \lambda \left( A\right) |A\subseteq M, A closed\right\} $
Later in the script the author writes, that we can deduce
Every Lebesgue measurable set can be written as the union of a Borel set and a null set.
from the first quote.
I have tried to proof this but I think I need help.