TITLE: Simple explanation for a Borel Set
QUESTION [7 upvotes]: I am from engineering background and the concept of Borel set, Borel field and measures sound  abstract to me. Can some one please, explain them in a simplified way i.e., to explain them without assuming the reader has any knowledge about measure theory.
For example when I read the function is continuous over some interval, I can visualize it and I know I can integrate it over that interval. But when I read X is the intensity measure for a Borel set, I don't get it and most importantly I don't get what should this imply for the future steps (i.e., if it is a Borel what can I do and if it is not what are the things that I can't do).
Assuming some one will answer these, can you please explain the Lebeg measure in the same manner simplified manner as well.
Hopefully this makes sense.

REPLY [7 votes]: When developing a theory of integration, one wants (roughly speaking) a somehow 'friendly' collection of sets over which a notion of 'size' (read: measure) can be defined. The first naive guess at what sets we can assign such a 'size' to is simply all sets in the space, but there are various sets that one can construct such that any 'nice' description of their 'size' will yield contradictions. 
The second naive guess would be the topology in the space (i.e. open intervals/closed intervals, arbitrary (finite) unions of open (closed) intervals arbitrary (finite) intersections of closed (open) intervals etc). It turns out, however that this is FAR too restrictive! There are all sorts of sets that ones' intuition suggests have a well defined size that are not open or closed. For example, one simply knows that $[0,1)$ should have 'size' equal 1, but if one is only considering open and closed sets, then this set will not a defined 'size'.
So one needs something between all sets, and the topological sets. The Borel sets sit in between these two naive guesses, and are the first step towards defining the right notion of sets with a legitimate 'size'.