 for all complex-valued Lebesgue integrable functions \(f\) almost everywhere, meaning for almost all initial conditions. The qualification ‘almost everywhere’ is non-trivial and is the source of a famous problem in the foundations of statistical mechanics, the so-called ‘measure zero problem’ (to which we turn in Section 3). So it is worth unpacking carefully what this condition involves. Not all sets have a finite size. In fact, there are sets of measure zero. This may sound abstract but is very natural. Take a ruler and measure the length of certain objects. You will find, for instance, that your pencil is 17cm long—in the language of mathematics this means that the one dimensional Lebegue measure of the pencil is 17. Now measure a geometrical point and answer the question: how long is the point? The answer is that such a point has no extension and so its length is zero. In mathematical parlance: a set consisting of a geometrical point is a measure zero set. The same goes for a set of two geometrical points: also two geometrical points together have no extension and hence have measure zero. Another example is the following: you have device to measure the surface of objects in a plane. You find out that an A4 sheet has a surface of 623.7 square centimetres. Then you are asked what the surface of a line is. The answer is: zero. Lines don’t have surfaces. So with respect to the two dimensional Lebesgue measure lines are measure zero sets.