re convergence properties, i have even found a source where the lebesgue integral is defined by this property.
i.e. a function is lebesgue integrable if and only if it is almost everywhere the pointwise limit of a sequence of (measurable) step functions, whose integrals converge to some number.
then the inetgeral is proved to be independent of thes equence of step functions and to depend only on the a.e. pointwise limit, and we're off.
have a good day!