(adsbygoogle = window.adsbygoogle || []).push({}); integral more general than the Lebesgue integral?

The Lebesgue integral is defined for measurable functions. But isn't it possible to define a more general integral defined for a larger class of functions?

I guess that we would then loose some of the fine properties of the Lebesgue integral - but which and why?

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# Integral more general then Lebesgue integral?

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