Why use measure theory for probability? (English)

...There are a million introductions to measure-theoretic probability out there, but I wanted to give one to address the complaints that many people (including myself) have, namely "Why should we care?" and "Why are sigma algebras and measures defined the way they are?"...

This lecture explains the reasons why we use the language of measure theory to do probability theory. The key point is that the undergraduate notions of probability density function (p.d.f.) and cumulative density function (c.d.f) are "density focused" and aren't so useful when considering more general probabilistic objects such as randomly generated functions. I then go on to give a brief introduction to probability measures.

http://www.youtube.com/watch?v=RjPXfUT7Odo