Null Set

Definition §

A set $E\subset \mathbb{R}$ is a null set (a set of measure zero) if for arbitrarily small $\epsilon >0$, there is some set of intervals, $(a_n,b_n)\subset \mathbb{R}$ where the following conditions hold:

1. $E\subset \bigcup _{n=1}^{\infty }(a_n,b_n)$
2. $\sum _{n=1}^{\infty }(b_n-a_n)<\epsilon$

Properties §

1. $\left|E_n\right|=0⟹\left|\bigcup _n{E}_n\right|=0$
2. $|E|=\{x\}$, a set containing a single point, is a null set
3. $|E|=\{x{\}}_{n=1}^{\infty }$, a countable set, is a null set (uncountable null sets exist too, e.g. Cantor’s set)

Examples §

1. Consider $f(x)=\{\begin{array}{l}1,\text{ if }x\in \mathbb{Q}\\ 0,\text{ if }x\notin \mathbb{Q}\end{array}$ . In this case, $f(x)=0$ a.e., since $\mathbb{Q}$ is countable.