1.1. Prologue: The problem of measure 3 Of course, one can point to the infinite (and uncountable) number of components in this disassembly as being the cause of this breakdown of intuition, and restrict attention to just finite partitions. But one still runs into trouble here for a number of reasons, the most striking of which is the Banach-Tarski paradox, which shows that the unit ball B := {(x, y, z) R3 : x2 + y2 + z2 1} in three dimensions2 can be disassembled into a finite number of pieces (in fact, just five pieces suffice), which can then be reassembled (after translating and rotating each of the pieces) to form two disjoint copies of the ball B. Here, the problem is that the pieces used in this decomposition are highly pathological in nature among other things, their construction requires use of the axiom of choice. (This is in fact necessary there are models of set theory without the axiom of choice in which the Banach-Tarski paradox does not occur, thanks to a famous theorem of Solovay [So1970].) Such pathological sets almost never come up in practical applications of mathematics. Because of this, the standard solution to the problem of measure has been to abandon the goal of measuring every subset E of Rd, and instead to settle for only measuring a certain subclass of “non-pathological” subsets of Rd, which are then referred to as the measurable sets. The problem of measure then divides into several subproblems: (i) What does it mean for a subset E of Rd to be measurable? (ii) If a set E is measurable, how does one define its measure? (iii) What nice properties or axioms does measure (or the concept of measurability) obey? (iv) Are “ordinary” sets such as cubes, balls, polyhedra, etc., measur- able? (v) Does the measure of an “ordinary” set equal the “naive geometric measure” of such sets? (For example, is the measure of an a × b rectangle equal to ab?) These questions are somewhat open-ended in formulation, and there is no unique answer to them in particular, one can expand the class of mea- surable sets at the expense of losing one or more nice properties of measure in the process (e.g. finite or countable additivity, translation invariance, or rotation invariance). However, there are two basic answers which, between them, suffice for most applications. The first is the concept of Jordan mea- sure (or Jordan content) of a Jordan measurable set, which is a concept closely related to that of the Riemann integral (or Darboux integral). This 2The paradox only works in three dimensions and higher, for reasons having to do with the group-theoretic property of amenability see §2.2 of An epsilon of room, Vol. I for further discussion.
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