Is the collection of Lebesgue measurable set the smallest $\sigma$-algebra containing sets with outer measure $0$ and the Borel $\sigma$-algebra?

Question

I am a student of Mathematics.I am self-studying measure theory.I have defined measurable set with respect to an outer measure $m^*$ as a set $A$ such that $m^*(E)=m^*(E\cap A)+m^*(E\cap A^c)$ for each set $E$.Measurable set with respect to the Lebesgue outer measure are called Lebesgue measurable sets.We know Borel sets are Lebesgue measurable and also know that sets with outer measure zero are Lebesgue measurable.Now I want to know if Lebesgue measurable sets is the smallest $\sigma$-algebra containing Borel sets and sets with outer measure $0$ and if so,how to prove it?Can someone help me with this? And if this is so,then is it true that Lebesgue measurable sets are generated by taking set difference between Borel sets and outer measure zero sets?