Lebesgue sigma algebra?

Question

Can anyone tell me strictly what Lebesgue sigma algebra is? Is it sigma algebra $m=\left\{ E \subset X | (\forall A \subset X) \mu ^{*} (A) = \mu ^{*} (A \cap E) + \mu ^{*} (A \cup E^c) \right\}$ on $X$, where $X$ is a set and $\mu ^{*}$ is the outer measure, from Caratheorody's theorem? Thank you, I am new to this

Answer 1

"Yes" under the extra conditions that $X=\mathbb R$ and $\mu^*$ denotes the outer Lebesgue measure (which is usually denoted as $\lambda^*$).

Answer 2

An equivalent definition is the sets $S \subseteq \mathbb{R}$ such that there exists a Borel set $B$ and a Borel set $N$ such that $\mu(N) = 0$ ($\mu$ being the Lebesgue measure) such that $S \triangle B \subseteq N$, where $\triangle$ is the symmetric difference, i.e. $S \triangle B = (S \setminus B) \cup (B \setminus S)$.