I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of cardinals, and never had a formal introduction to Set theory or Cardinals theory. The concept of PTI was amusing to me so I […]
If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of a countable collection $(I_j)$ of half-open intervals then $\sum_j|I_j|=1$. It’s easy to conclude that this is easier than it really […]
I know what transfinite induction is, but not sure how it is used to prove something. Can anyone show how transfinite induction is used to prove something? A simple case is OK.