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I study physics and in this year I have to formule and write my bachelor thesis. I have a lot of ideas but some of them looks more interesting for me.

A few days ago I thought about situations in physics where Riemman Integral is not sufficient and my idea is to write about parts of physics in which Lebesgue Integral is needed. Unfortunately, in my courses during my studies, no one shows me situations in physics where Riemman integral is not enough so I ask you, Maybe someone of you knows examples in physics where Lebesgue Integral is essential ? After reading about Lebesgue integral, I tried to find informations about measure theory in physics but I didn't find nothing interesting.

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A very easy answer is Quantum Mechanics. For example, the space of states of the harmonic oscillator is $L^2(\mathbb{R})$, the space of square (Lebesgue) integrable functions.
Also, Lebesgue integration is used to properly define the Fourier Transform, which is, as you probably know by now, an essential tool in physics.

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