Does Lebesgue integral satisfy Riemann integral properties? From what I did study, I know that Lebesgue integral is more general than Riemann integral.
Then, does Lebesgue integral satisfy all of the Riemann integral properties?
In particular, is the following true?
For some given set X,
$\int_{A}f d\mu + \int_{B}f d\mu = \int_{A\cup B}f d\mu$ 
where $M$ is a $\sigma$-algebra in a set $X$ and $\mu$ is a positive measure on $M$.

As was stated in the comments, if the domain is compact then the Riemann and Lebesgue integral agree.  But one thing Riemann has over Lebesgue is that it allows improper integrals.  This question from 2013 gives an example:
Riemann-integrable (improperly) but not Lebesgue-integrable
