Littlewood’s three principles (1)

by tzy9393

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}