The Collection of Lebesgue Measurable Sets

The Collection of Lebesgue Measurable Sets

We have already looked at whether certain sets are Lebesgue measurable. We've already noted that $\emptyset$, $\mathbb{R}$, every countable set, and every interval is a Lebesgue measurable set. We now give a more general definition of the collection of Lebesgue measurable sets.

Theorem 1: Let $\mathcal M$ be the set of Lebesgue measurable sets. Then $\mathcal M$ is the $\sigma$-algebra that contains the $\sigma$-algebra of Borel sets and every interval, open set, $G_{\delta}$ set, and $F_{\sigma}$ set is contained in $\mathcal M$.
  • Proof: We have already established that $\mathcal M$ is a $\sigma$-algebra. Let $O$ be any open set in $\mathbb{R}$. Since $O$ is open, there exists a countable collection of disjoint open intervals $\{ I_n : n \in \mathbb{N} \}$ such that $\displaystyle{O = \bigcup_{n=1}^{\infty} I_n}$. Since $I_n \in \mathcal M$ for all $n \in \mathbb{N}$, we have that $O \in \mathcal M$. The $\sigma$-algebra of Borel sets is the smallest algebra containing all the open sets of $\mathbb{R}$. Therefore, $\mathcal M$ contains the $\sigma$-algebra of Borel sets.
  • We have already established that $I \in \mathcal M$ for every interval $I \in \mathcal P (\mathbb{R})$.
  • Let $G$ be a $G_{\delta}$-set. Then $G$ is a countable intersection of open sets, say $\displaystyle{G = \bigcap_{n=1}^{\infty} O_n}$. Since each $O_n \in \mathcal M$ we have that $G \in \mathcal M$.
  • Lastly, let $F$ be an $F_{\sigma}$-set. Then $F$ is a countable union of closed sets, say $\displaystyle{F = \bigcup_{n=1}^{\infty} F_n}$. Since each $F_n^c$ is open, $F_n^c \in \mathcal M$. So $F_n = (F_n^c)^c \in \mathcal M$ and $F$ is a countable union of Lebesgue measurable sets so $F \in \mathcal M$. $\blacksquare$
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