Usually, random variable is defined as a measurable function, with respect to Borel sigma-algebra on real numbers. Why? What would happen if we replace Borel by Lebesgue? Could you give an example of a statement in probability theory, which would not be true for random variables defined with respect sigma algebra of Lebesgue measurable sets.
I always had feeling that we are fine till we are allowed to integrate.
First of all, if we require random variable to be measurable with respect to Lebesgue sigma algebra, this would be STRONGER constrain. So there are no general statements, which would become wrong. What actually will happen is that we will lose many examples of random variables (not all continuous functions $X \colon \mathbb{R} \rightarrow \mathbb{R}$ would be good enough).