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
2026-05-05 21:30:06.1778016606
Measure Theory ; why it works?
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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:
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:
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.