 From a more formal point of view, a measure assigns numbers to certain subsets of a set \(X\) (see Appendix B for a formal definition). This can be done in different ways and hence there are different measures. Consider the example of a plane. There is a measure that simply assigns to each appropriate region of a plane the area of that region. But now imagine that we pour a bucket of sugar on the plane. The sugar is not evenly distributed; there are little heaps in some places while there is almost no sugar in other places. A measure different from the area measure is one that assigns to a region a number that is equal to the amount of sugar on that region. One of these measures is particularly important, namely the so-called Lebesgue measure. This measure has an intuitive interpretation: it is just a precise formalisation of the measure we commonly use in geometry. The interval [0, 2] has Lebesgue measure 2 and the interval [3, 4] has Lebegues measure 1. In two dimensions, a square whose sides have Lebesgue measure 2 has Lebesgue measure 4; etc. Although this sounds simple, the mathematical theory of measures is rather involved. We state the basics of measure theory in the  Appendix,  Section B, and avoid appeal to technical issues in measure theory in what follows.