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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.

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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'.

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This is only a partial answer to your question, but I think it's important. A large part of the reason for introducing the Lesbesgue measure and Lesbegue integral is taking limits. Riemann integrable functions are not closed under taking limits. For example, if $\{f_n\}\to f$ is a sequence of Riemann integrable functions, then the limit is not necessarily Riemann integrable. However, if the $f_n$'s are all measurable, then we can conclude that $f$ is measurable (which is basically the same as saying Lesbesgue integrable).

It is natural to want to say things like $\lim_{n\to\infty}\int f_n=\int f$. Obviously this is not always the case, but the point is that we need much weaker assumptions on the functions $f_n$ and the manner in which they converge to $f$ in order to make such a statement using the Lesbegue integral than we would need to make this statement using the Riemann integral. For this statement to hold with Riemann integrable functions, we must assume $\{f_n\}\to f$ unifomly, while for Lesbegue integrable functions, we can almost get away with assuming only pointwise convergence. See the dominated and monotone convergence theorems.

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  • $\begingroup$ Thank you very much. This pointed out the need and the importance of Lesbeg integrable functions. $\endgroup$ – OAH Dec 10 '13 at 20:43

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