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I am trying to appreciate how the better properties of the Lebesgue integral make it more useful than the Riemann integral for studying things like Fourier Analysis or PDEs. I've seen how the Lebesgue integral behaves better under taking pointwise limits and how this is a result of the Lebesgue measure being closed under taking countable unions.

When reading about this, one example that is given is the function $f=\sum_n f_n$ where $f_n=(r_n-\epsilon/2^n,r_n+\epsilon/2^n)$ and $r_n$ is an ennumeration of the rationals in the unit interval. This can yield an example of an increasing sequence of integrable functions whose limit is not Riemann integrable.

Examples like these are nice, but they also seem pathological. Could anyone provide an application or particular theorem where the properties of the Lebesgue integral are indispensable and not just used to integrate a function that's not Riemann integrable or simplify a proof that would be messier without them?

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"..not just used to integrate a function that's not Riemann integrable"

But that is a major advantage. The class of functions that are Lebesgue-Integrable is significantly greater than those that are Riemann-Integrable.

For an example of a speciic Theorem, consider Lebesgue's monotone convergence theorem or the Dominated Convergance Theorem, both based on Lebesgue Integrals.

EDIT

There is a great MSM post, titled Lebesgue integral basics , that I believe you will find very enlightening.

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