This is a question from Real Analysis by Royden, 4th edition. (#5, pg. 34)
Using properties of outer measure, prove that $[0,1]$ is uncountable.
I believe that I am going to have to assume otherwise and use countable subadditivity in some way to produce a contradiction. I am not sure how to do so though. What other properties of outer measure might I need?
Thank you for any help!
Best Answer
Do you already know points have measure zero?
If so, then enumerate all points in $[0,1]$. The "countable" union of all these points is the entire interval, with measure $1$. The "countable" sum of the measures of all these points, on the other hand, is...