Lebesgue sigma algebra

Question

A set $E\subseteq \mathbb{R}$ is called $m^*$-measurable if for all $A\subseteq \mathbb{R}$ $$m^*(A)=m^*(A\cap E)+m^*(A\cap E^c)$$

The set of all measurable sets is called Lebesgue sigma algebra

Does not sigma algebra has properties? this is a way to "build" Lebesgue sigma algebra from "another way" by measurable sets?

Answer 1

Yes, a $\sigma$-algebra has properties ! Let

$ \mathcal{L}= \{E \subseteq \mathbb R: m^*(A)=m^*(A\cap E)+m^*(A\cap E^c) \quad \forall A \subseteq \mathbb R\}.$

$ \mathcal{L}$ has the following properties (try a proof):

1. $ \mathbb R \in \mathcal{L}$;

2. $E\in \mathcal{L}$ implies $\mathbb R \setminus E \in \mathcal{L}$;

3. if $(E_j)$ is a sequence in $ \mathcal{L}$, then $\bigcup E_j \in \mathcal{L}.$

This shows that $ \mathcal{L}$ is a $ \sigma$-algebra.