A $\sigma$-algebra $\mathcal{S}$ on $\mathbb{R}$ that is larger than $\mathcal{L}$ s.t. leb. outer measure is a measure on $(\mathbb{R},\mathcal{S})$.

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The author wrote as follows on p.52 in this book:

We have accomplished the major goal of this section, which was to show that outer measure restricted to Borel sets is a measure. As we will see in this subsection, outer measure is actually a measure on a somewhat larger class of sets called the Lebesgue measurable sets.

Is there a $\sigma$-algebra $\mathcal{S}$ on $\mathbb{R}$ such that the lebesgue outer measure $\lambda^*$ (https://en.wikipedia.org/wiki/Lebesgue_measure) is a measure on $(\mathbb{R},\mathcal{S})$ and $\mathcal{L}\subsetneq\mathcal{S}$, where $\mathcal{L}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb{R}$?