The difference between Riemann integrable function and Lebesgue integrable function

My professor asked me how to intuitively understand Lebesgue's dominated convergence theorem and what's the effect of the integrable dominated function. More specifically, when we are given a Lebesgue integrable function, why it suffices to consider other things on a subset with finite measure?

I think the professor means the following theorem：

If $f$ is Lebesgue integrate in $X$, for every $\epsilon>0$, there exists measurable $E$ with $\mu(E)<\infty$ s.t. $\int_{X\setminus E}|f|<\epsilon$.

Hence I proved the statement using monotone convergence of $|f|\chi_{B(x_0,n)}$, but it seems my professor was not satisfied. She wanted me to explain what's the essential difference between a Lebesgue integrable function and a Riemann integrable function, and what makes a Lebesgue integrable function not Riemann integrable. She said answering the second question will help me answer the original question.

I think from the definition of Riemann integral and the theorem, the Lebesgue integrable function is Riemann integrable if and only if it's continuous, i.e. Riemann integrability requires the function value to not have a large oscillation when $x$ changes slightly. Is it a correct explanation? How about Lebesgue integral? And how is it related to the first question?

I know this question may be a little subjective and may not have the so-called "correct" answer as other questions on the website. But I am still looking forward to your comments.

• So many views, so little activity. A dead, but seemingly good post. – Simply Beautiful Art Dec 31 '16 at 21:47