Let $I$ an interval on $\mathbb{R}$ such as $I=(a,b)$, with $a$ or $b$ could be equal to infinity.
And we have $f\in \mathcal{L}^1(I,\mathcal{B}(I), \lambda)$, then do we have always $$\int_{(a,b)}fd\lambda= \int_a^bf(x)dx$$and if not ! When we have this equality, knowing that $f\in \mathcal{L}^1(I,\mathcal{B}(I), \lambda)$?
Take simply the characteristic function of the rationals $f = \chi_{\mathbb Q}$. $f$ is Lebesgue-integrable (and $\int fd\lambda$ = 0) but $f$ is not Riemann-integrable.
In the case of bounded functions over closed intervals, a Lebesgue-integrable function $f$ is Riemann-integrable iff the set of discontinuities of $f$ is a null set.