Helllo I'm Studying measure theory ( Lebesgue and Fatou .. ) in University but I don't understand the utility of it ? i don't find any motivation to study it ; it seems complicated and i don't find good books show me it's beauty so : Can some one give me some reasons as motivation to it ? and Suggest me a good books in the kind of Serawy books in Physics but in Topology and Measure Theory ? and Thank you very Much
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The most basic reason why measure theory is needed is as follows.
When trying to define the integral of a function, one approach is to use a Riemann sum. Say our function is $f$, defined on an interval $I$. In the Riemann sum approach, we approximate the integral of $f$ by chopping $I$ into a finite number of pieces, and then saying that the integral of $f$ on each piece is equal to the length of that piece times some number which approximates the height of $f$ on that subinterval. A Riemann sum looks like this:
(Length of subinterval $I_1$) $\times$ (Height of $f$ on $I_1$) + ...
Lebesgue had an idea for a different way of defining the integral, based on chopping up the range of the function instead of the domain. Sometimes the function $f$ is between $0$ and $1$, other times it's between $1$ and $2$. If we take the length of the set on which $f$ is between $1$ and $2$, and then multiply that by, say, $2$, we get a rough upper bound for the integral of the function on that set. So a Lebesgue integral looks like this:
(Some height $h$)*(Length of $\{x\in I\mid f(x) \text{ is near $h$}\}$) +...
The trouble is that this definition leads us to calculate the length (measure) of sets of the form $f^{-1}(J)$, where $J$ is an interval. If $f$ is sufficiently weird, there's no telling how bizarre that set could be. Can we even be sure it has a well-defined length? Thus a lot of measure theory is devoted to defining the concept of length carefully, and working out for which functions $f$ we can guarantee that these sets will have a well-defined length - these are called measurable functions.
Jack M
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That answer would be valid if there were no other way to define the Lebesgue integral. (There are lots of different ways that do not use measure theory). The best you can say here is that Lebesgue found that measure theory was a useful tool for developing an integral more general than the Riemann integral. The other comment about "tools" in general is better I think. There is still room here for an answer that is more compelling. – B. S. Thomson Nov 03 '15 at 18:13
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@B.S.Thomson Well, let's just say that measure theory is one of the most obvious and natural approaches to defining Lebesgue's integral. Once you realize you're going to need to know the length of $f^{-1}(I)$ it's natural to start trying to work out a general theory of which subsets of $\mathbb R$ have a notion of length so that you can work out for which $f$ your theory applies. In any case, I think this is why measure theory was originally developped. – Jack M Nov 03 '15 at 18:41
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I disagree. It is both "obvious and natural" at your level of mathematical maturity. At that level you hardly distinguish between a measure and an integral and, indeed, should not. But for a student now and for a great many mathematicians in 1901 this trick of Lebesgue's of developing an integral by using measure theory was both subtle and unnatural. If he had defined his integral using Riemann sums [as he could have but did not] everyone would have said that it was "a natural and obvious" generalization of the traditional integral. – B. S. Thomson Nov 04 '15 at 21:25
In the context of real analysis, it is not possible to use the "standard" Riemann integral for all kinds of functions, and often we want to use the more "powerful" Lebesgue integral, which is often derived using measure theory.
– Olorun Nov 03 '15 at 08:11