Littlewood's First Principle

Littlewood stated the principles in his 1944 Lectures on the Theory of Functions

Approximation by Compact Sets

The main result is the following: Let </math> E </math> be a Borel measurable set. For every </math>\varepsilon > 0</math> there exists an open set </math> U </math> with </math> E\subset U </math> such that </math> m(E)>m(U)-\varepsilon </math>. Note that this implies </math> 0< m(U)-m(E)<\varepsilon </math> by monotonicity.

Proof: First suppose </math> E </math> is bounded. If </math> E </math> is also closed, then we're done; </math> E </math> is compact. Otherwise, the boundary </math> \partial{E}=\overline{E}\backslash E </math> is nonempty. Now, </math> \partial{E} </math> is a Borel measurable set, so we can approximate it from above by the previous section. That is, there exists an open set </math> U\supset \partial{E} </math> such that </math> m(U) < m(\overline{E}\backslash E) +\varepsilon = \varepsilon </math>. Now define </math> K= \overline{E} \backslash U </math>. By construction, this set is contained in </math> U </math>. Moreover, it is closed, being the intersection of two closed sets. Finally, </math> K </math> is bounded because </math> E </math> is bounded. By the Heine-Borel Theorem, </math> K </math> is compact, and </math> K </math> satisfies the constraints of the theorem because

$$m(K)=m({\overline {E}}\backslash U)=m(E)-M(E\cap U)=m(E)-m(U)+m(U/E)\geq m(E)-m(U)+m({\overline {E}}\backslash E)\geq m(E)-\varepsilon$$

Now assume </math> E </math> is unbounded. Let </math> E_j = E \cap [j, j+1) </math>. By the preceding argument, for each </math> \varepsilon > 0 </math> there exist compact </math> K_j </math> such that </math> K_j \subset E_j </math> and </math> m(K_j) > m(E_j)-\varepsilon / 2^k </math>. Let </math> H_n = \bigcup_{j=-n}^{n} K_j </math>. Then </math> H_n </math> is compact and </math> m(H_n) \geq m(\bigcup_{j=-n}^n E_j) + \varepsilon </math>, and since </math> \lim_{n\to\infty} m(E)=m(\bigcup_{j=-n}^n E_j) </math>, the result follows.

Littlewood's First Principle

The previous two results imply the following formulation of Littlewood's First Principle: If </math> E </math> is a Borel measurable set and </math> m(E)<+\infty </math>, then for every </math> \varepsilon > 0</math> there is a set </math> A </math> that is a finite union of open intervals such that </math> m(A\delta E) < \varepsilon </math>, where </math> \delta </math> denotes symmetric difference.

Proof: Using the previous two sections find </math> K </math> and </math> U </math> such that </math> K\subset E \subset U </math> and </math> m(E)-\epsilon < m(K) </math> and </math> m(E)+\epsilon > m(U) </math>. As noted in the proof of the first section, </math> U </math> is really a countable union of open intervals. These intervals form an open cover for </math> K </math> and since </math> K </math> is compact, only finitely many cover </math> K </math>. Defining </math> A </math> to be the union of these finitely many intervals, the result follows, since </math> m(A/E) \leq m(U/E) </math> and </math> m(E/A) \leq m(E/K) </math>.