I think the problem to grapple with is that I can cover the rationals in [0,1] with countably many intervals of total length only 1⁄2 (eg enumerate rationals in [0,1], and place interval of length 1⁄4 around first rational, interval of length 1⁄8 around the second, etc). This is not possible with reals—that’s the insight that makes measure theory work!
The covering means that the rationals in an interval cannot have a well defined length or measure which behaves reasonably under countable unions. This is a big barrier to doing probability theory. The same problem happens with ANY countable set—the reals only avoid it by being uncountable.
I think the problem to grapple with is that I can cover the rationals in [0,1] with countably many intervals of total length only 1⁄2 (eg enumerate rationals in [0,1], and place interval of length 1⁄4 around first rational, interval of length 1⁄8 around the second, etc). This is not possible with reals—that’s the insight that makes measure theory work!
The covering means that the rationals in an interval cannot have a well defined length or measure which behaves reasonably under countable unions. This is a big barrier to doing probability theory. The same problem happens with ANY countable set—the reals only avoid it by being uncountable.