### Littlewood’s three principles (1)

Littlewood principles describe the relationship between concepts in measure space and concepts in topological space. He summerized these connections which provide a quite intuitive guide:

• Every set is nearly a finite union of intervals.
• Every function is nearly continuous.
• EVery convergent sequence is nearly uniformly convergent.

However, in mathematics we need a precise meaning of the word “nearly”. So let us investigate them in each case one by one. First off, let us introduce the notation we will use. ${m(E)}$ denotes Lebesgue measure of a measurable subset ${E}$ in ${\mathbb{R}^{d}}$, where we define Lebesgue measurable in the sense that for every ${\epsilon}$, there exists an open set ${O}$ such that ${E\subseteq O}$ and ${m_{*}(O-E)\leq\epsilon}$ and for Lebesgue measurable subset ${E}$, ${m(E)=m_{*}(E)}$. Here ${m_{*}(E)}$ denotes the Lebesgue outer measure of a measurable subset which is defined to be ${\mathrm{inf}\sum_{i=1}^{\infty}\left|Q_{i}\right|}$ and the infimum is taken over all countable coverings of ${E}$ by closed cubes.

Theorem 1 Suppoe E is a measurable subset of ${\mathbb{R}^{d}}$ with finite measure. Then, for every ${\epsilon>0}$, there exists a finite union ${F=\bigcup_{i=1}^{N}Q_{i}}$ of closed cubes such that ${m(E\triangle F)\leq\epsilon}$.

Proof: Fix ${\epsilon>0}$. By definition, for this ${\epsilon}$, we are able to choose a family of closed cubes ${\left\{Q_{i}\right\}_{i=1}^{\infty}}$ such that ${E\subseteq\bigcup_{i=1}^{\infty}Q_{i}}$ and ${\sum_{i=1}^{\infty}\left|Q_{i}\right|\leq m(E)+\epsilon}$. Since we assume ${m(E)<\infty}$, the series on the left hand side converges, i.e., there exists ${N>0}$ such that ${\sum_{i>N}\left|Q_{i}\right|<\epsilon}$. Now we let ${F=\bigcup_{i=1}^{N}Q_{i}}$, then

$\displaystyle \begin{array}{lll} m(E\triangle F) & = & m(E-F)+m(F-E)\\ & \leq & m(\bigcup_{i=N+1}^{\infty}Q_{i})+m(\bigcup_{i=1}^{\infty}Q_{i}-E)\\ & \leq & \sum_{i>N}\left|Q_{i}\right|+\sum_{i=1}^{\infty}\left|Q_{i}\right|-m(E)\\ &\leq& 2\epsilon \end{array}$