I think this becomes much more intuitive once one understands that the cardinality of any nondegenerate closed interval is the same as the cardinality of the continuum.

It becomes perhaps even more obvious when you realize that any non-empty open interval (a,b) with a<b is homeomorphic to the reals. Once that really sinks in, much of this becomes easier.

> the cardinality of any closed interval is the same as the cardinality of the continuum.

In English: There are just as many real numbers (rational and irrational combined) in between any two real numbers as there are real numbers total. (That's a somewhat simplified statement as I didn't really define 'closed interval', but it serves to give a taste of the concept.)

If nothing else, statements like those completely break your intuitive notions regarding size in this context and convince you that you can only proceed based on logic and proven theorems. Which is, of course, accurate, at least until you develop a new intuition.

Measure theory is the branch of mathematics concerned with developing new notions of size appropriate to different contexts.

http://en.wikipedia.org/wiki/Measure_(mathematics)

> Measure theory is the branch of mathematics concerned with developing new notions of size appropriate to different contexts.

While this is probably among the most interesting definitions I've ever seen, I'm not sure it is correct. Cardinality, as we've discussed it here, is usually considered to be in the domain of set theory; measure theory assigns measures in a multitude of interesting ways, but, at least in all of it with which I'm familiar, those measures come only from [0, ∞], not from some more exotic domain of values. (A lot of the measure-theory proofs that I know do rely in this being the set of values of a measure—for example, on being able to subtract most of the time, and on having only one kind of infinite measure—so just saying "let's allow different values" doesn't immediately rescue it.)