TITLE: General condition that Riemann and Lebesgue integrals are the same
QUESTION [18 upvotes]: I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two integrals are the same. 
However, I want to know as general as possible condition for two integrals to be the same. For example, in terms of improper integrals, unbounded functions and R^n space.
Any answer would be helpful. Thank you.

REPLY [6 votes]: Your statement about "a bounded Riemann integrable function" is correct, but you don't need the "bounded" part: an unbounded function can't be Riemann integrable.
It can have an improper Riemann integral, but that's a different matter.
Lebesgue's criterion for Riemann integrability: A function on a bounded interval $I$ is Riemann integrable if and only if it is 

bounded, and 
continuous almost everywhere (i.e. except on a set of Lebesgue measure $0$).