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Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a question here. Why do we need such a general concept of measure? Length is a kind of measure, but why did Lebesgue think of so general a concept of "measure" when he just needed the length to evaluate his integrals? Is there any other motivation behind this ?

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    $\begingroup$ Good question. A couple of random comments. 1. Historically, measure theory came first. 2. To define Riemann integration in one dimension, all we need is a definition of the length of an interval. But to define Lebesgue integration, we need a definition of the length of potentially much more complicated sets. $\endgroup$ – goblin Jul 7 '13 at 6:44
  • $\begingroup$ "Why do we need such a general concept of measure?" I mean, isn't anything worth having worth generalizing? Wouldn't it be cool to be able to talk about the lengths of arbitrary (reasonable) subsets of $\mathbb{R}$, in principle -- or at least to generalize past finite unions of intervals, in practice? And how cool would it be if moreover we could describe length (i.e. measure) axiomatically? $\endgroup$ – Jesse Madnick Jul 7 '13 at 9:16
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    $\begingroup$ Well I used to think measure theory is Lebesgue integration theory... $\endgroup$ – Hui Yu Jul 7 '13 at 12:41
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I'm not sure you can say there's a single motivation behind introducing a concept, but I found multiple motivations behind the introduction to measure theory I saw.

First, the Cantor set. It's hard to talk about the length of the Cantor set rigorously since it's not a single interval, but the result of a countable intersection of unions of intervals. Lebesgue measure makes it easier to see that the "length" of the Cantor set is zero, and then to see that length and cardinality are not very well correlated.

Second, once Lebesgue measure is defined for $\mathbb{R}^d$, integration in multiple dimensions is essentially as easily defined as integration in one dimension (see Stein and Shakarchi's Real Analysis where the Lebesgue integral is immediately defined for arbitrary positive integer dimension). Another matter brought up in that book is that not all possible Fourier series, in the sense of sequences $\{ a_n \} \in \ell^2(\mathbb{Z})$, have corresponding Riemann integrable functions. However, they do have corresponding Lebesgue integrable functions, and this integral requires the Lebesgue measure.

But for more complicated measures, I think the main motivations I saw in my case were probability spaces (which leads to cool applications in probability), and the counting measure. One of the really nice things about the counting measure is that it clearly shows that sums and integrals are "the same", so that you can prove theorems about discrete and continuous objects at the same time (for example, Hölder's, Minkowski's, and Hilbert's inequalities).

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