# Lebesgue sigma algebra

A set $$E\subseteq \mathbb{R}$$ is called 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?

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 follwing 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

• A sigma algebra is closed to any countable unions, not necessary disjoint. – Mark Feb 18 at 10:42
• So why did my lecture said that if Carathéodory criterion is met, it is a Lebesgue sigma algebra, is should be just a sigma algebra – gbox Feb 18 at 10:47