TITLE: Understanding Borel sets
QUESTION [75 upvotes]: I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just need more examples. 
Also I want to understand what is Borel $\sigma$-algebra.

REPLY [7 votes]: Another strange Borel set: the set of all numbers in [0,1] whose decimal expansion does not contain 7. 
One more: The set of all real numbers in [0,1] whose decimal expansion contains 2 or 5. (Cantor type set)
Last but not least: The set of all real numbers in [0,1] whose decimal expansion contains only finitely many 6.