Do physicists in general have need of the Lebesgue integral? Are they taught it in graduate school?
I'm not a physicist so I don't know. I'm told that Lebesgue measure is very useful for probability theory so I'm sure there's lots of times when it will be very useful for physics.
Mathematical physicists: Yes.
Theoretical physicists: Maybe.
Experimental physicists: No.
Mathematical physicists need it mainly to understand proofs in books on functional analysis.
As a disclaimer, I don't know anything about physics. However, I would guess that it would depend on the nature of things you will have to integrate. Many physical things, objects and natural-enough shape-esque objects will be sufficiently integrated with Riemann integration, but if you have to integrate more complicated entities, you will need the Lebesgue integral.
To gaussmouse and wisvuze: impressive that you both feel qualified to answer a question about physicists when you both claim you know nothing about physics.
As a mathematician I will say that Lebesgue integration is needed to prove theorems concerning integrals and related material. For applications it is only necessary to know what limitations the theory may impose.
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