From: Bryan VanDeVen Subject: Re: help me with this paradox Date: Fri, 20 Oct 2000 17:29:20 -0500 Newsgroups: sci.math Summary: [missing] > What is measure theory? (Please try to keep it simple) Usually a graduate course, in most math depts. :) Measure theory has do to with how we measure the "size" of sets. In turn it has to do with how we integrate. Since probability, in essence, is concerned with nothing more than integration (of density functions), it has ramifications for probability as well. Now, as you may (or may not) know, the integral of a function at a single point has a value of zero (because a point has "measure zero") That is why, in your example, choosing a single random number from [0,2Pi), the probability is zero. The probability of picking 1.2 is the integral of 1/2Pi from 1.2 to 1.2 - just a single point - which is zero. "But you can point north!" you say. Well, probability is a _model_, not real life. Very important distinction. You cannot, in real life, move less than some distance (the planck distance maybe?) so "north" _must_ be defined as an interval "about north" and thus the probablity (as computed above) will not be zero anymore (just incredible small). So the easy answer is that a uniform distribution is not a completely accurate model for picking random compass directions (in real life). Pick a better model, and everything works out like you want. If you are still bothered, that in an abstract sense, on some "ideal" compass, that the probability of picking one point out of the uncountably infinitely many points in an interval is zero, just consider that if it we defined it any other way, our useful system would break in ways that would make it not so useful. Also consider that working with infinite systems always makes issues more complex and subtle. Just because P(A)=0 means "A is impossible" when dealing with a finite number of things, does not mean that it has to mean exactly the same thing when we pass to an infinite system, because now P(A) is the result of an limiting process. But relax, there are even stranger things. Something called the "Axiom of Choice", which most all mathematicians accept, is equivalent to the existence of non-measurable sets - that is, there are events (sets) for which _no_ probability can even be computed. That is, there is a set E such that P(E) does not exist - not "is zero", but does not exist at all. Regards, -- Bryan Van de Ven Applied Research Labs University of Texas, Austin ============================================================================== From: Stephen Montgomery-Smith Subject: Re: help me with this paradox Date: Sat, 21 Oct 2000 20:38:59 GMT Newsgroups: sci.math jolt64@my-deja.com wrote: > > What is measure theory? (Please try to keep it simple) > It is kind of hard to give a simple answer. The idea is to lead to a very general theory of integration that integrates as many functions as is reasonably possible. So for example, Lebesgue measure is measure on the real line. For each set A that is a subset of the real line, you try to define a notion of length of real line in the following way: measure of [a,b] = b-a and if A_1, A_2, ... is a countable collection of disjoint sets, then measure of union of A_n = sum of measures of A_n. Well, you cannot get the measure of every set in this way - what you do is start with the intervals, then find all sets that can be reached by a process of taking countable unions, intersections and set complements (and then to finish it off add in subsets of sets of measure zero). You can get a lot of sets in this way, but (at least if you believe axiom of choice) not every set. Interesting are sets of measure zero. This includes, for example, the set of rational numbers - indeed all countable sets. But there are other sets, for example, the so called Cantor set, which consists of those numbers in [0,1] that can be written in base 3 using only 0's and 2's. Once you have a notion of measure, then you get a notion of integration. You start with so called simple functions, that is, functions that only take finitely many values. For these it is clear what their integral should be. Then you get other functions by taking various limits of these functions. It is actually a rather large subject, and to really begin to understand it you need to read a book or take a graduate math course on it. -- Stephen Montgomery-Smith stephen@math.missouri.edu http://www.math.missouri.edu/~stephen