|R| ≠ |R²| in probability theory

valjok

According to Cantor, the cardinality of a line is equal to cardinality of a plane; that is, both sets have equally many elements because there is 1-to-1 mapping between them. Based on his set theory, Kolmogorov developed a probability theory. In this theory, however, the probability of line outcome is equal to zero on plane probability space; that is, the continous plane has infinitely more points than a continous line in it. It is impossible that we get a line point when randomly pick up one from a given plane (throwing a dart example).

Do you see the contradiction |R| = |R²| vs. |R|/|R²| = 1/∞? Am I missing something?

HallsofIvy

I don't understand why you think those two completely different things should be the same. The measure of a (non-countable) set has little to do with its cardinality.

(I added "non-countable" because any countable set has cardinality 0- but still, there is no relation between cardinality and measure for non-countable sets.)

statdad

"I added "non-countable" because any countable set has cardinality 0- "

I believe you meant to say a countable set has measure 0

HallsofIvy

Yes, of course. Thank you.

valjok

Excuse me my ignorance, but I do not distinguish between the measure and cardinality. I consider the number of elements in two sets (R^2 and its R subset), which is cardinality. One theory tells that there is equal number of elements. Another gives more intuitive answer.

HallsofIvy

Then you should! The sets [0, 1] and [0, 2] have exactly the same cardinality but different measures.

valjok

Thank you. I have a clue but still do not understand the need for measure if all I need to compute the probability is the number of elements in a set (we were taught in the university that the number of elements in a set is called cardinality)? Now, I read a book on probability theory that puts it like on this site http://www.cut-the-knot.org/Probabil...ctionary.shtml . Just number of elements in a set is important and it tells nothing about the measures. Is the sigma-algebra the key?

Hurkyl

Defining a probability measure in terms of counting elements cannot give a reasonable answer if we want to talk about ideas like a uniform probability distribution on [0,1], and that a sample has a 50% chance of lying in the subinterval [0,1/2].

The idea that the interval [0,1/2] is "half" of the interval [0,1] is a geometric idea and has absolutely nothing to do with cardinality. If we want to define probabilities that relate to geometric ideas, we're probably going to have to use geometric methods in our probability theory.

Kolmogorov detailed a way to do so, and it worked.

And the neat trick is that we already know that the domain where "counting elements" works turns out to be a special case of these geometric methods. (A measure on a finite set turns out to be equivalent to simply assigning a nonnegative 'weight' to each element of that set)

HallsofIvy

The probability depends on the number of elements in a set only for finite sets!

Suppose you have a uniform probability distribution on [0, 4]. That is, you choose a real number from 0 to 4 and every such number is equally likely to be chosen.

The probability that the number chosen is in [0, 1] is (1- 0)/(4- 0)= 1/4. The probability that the number chosen is in [0, 2] is (2- 0)/(4- 0)= 1/2. But those two sets have exactly the same cardinality.

Again, the idea that the probability a number is in a given set depends on the cardinality of the set applies only to "discrete" probability where the sets are finite.

For infinite sets, you have to define the measure of the set- essentially by giving a probability distribution on the sets.

valjok

Thank you, guys. With this topic I wanted to clarify the notion of cardinality. Studying in university, I remember the amazement when lecturer told us the basic fact about set theory: there is 2^N more elements in [0,1] than in N. This is counterintuitive and now I see that matematitians also disagree with Cantor's element countng when entail the more adequate/accurate measure device.

Hurkyl

You're not being perfectly clear here, but I think you have the wrong idea. There is nothing wrong with Cantor's element counting -- it's just that lots of problems aren't counting problems.

valjok

Hurkyl, measuring the probability spaces is one of examples where more adequate element counting is needed. The presence of many other problems must not obstruct this need.

Disclaimer! I understand that there is something very deep in the Cantor's concept. The fact that the natural numbers have zero measure just proves this.