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. denotes Lebesgue measure of a measurable subset in , where we define Lebesgue measurable in the sense that for every , there exists an open set such that and and for Lebesgue measurable subset , . Here denotes the Lebesgue outer measure of a measurable subset which is defined to be and the infimum is taken over all countable coverings of by closed cubes.
Theorem 1 Suppoe E is a measurable subset of with finite measure. Then, for every , there exists a finite union of closed cubes such that .
Proof: Fix . By definition, for this , we are able to choose a family of closed cubes such that and . Since we assume , the series on the left hand side converges, i.e., there exists such that . Now we let , then