A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) definition of expectation; he said he is confused by "simple facts" like $$ E(c)=c, c\in \mathbb{R} $$ and why the expectation is linear.
Embarrassed that I did not explain this well enough earlier, I tried to explain to him expectation is a form of weighted sum, and for continuous case we are using some kind of probability measure coming from the density function. So the expectation is really some kind of integration. The student seemed to follow at this point. Then I tried to explain that the integration involved this way is not the same as the Riemann integration he learned in calculus classes, which is partly why the expectation of Cauchy distribution does not exist. I drew a graph and showed the Lebesgue integral can be viewed as a kind of "horizontal"-decomposition of the integration area. He asked me a very good question:
"What is the benefit of using horizontal instead of vertical? Isn't that the same thing?"
I did not really know how to answer this appropriately in a short time. I showed him that the horizontal decomposition would involve general (measurable) sets, not rectangles. And I also intuitively defined the outer measure of a set using open boxes. As a practice I showed that $\mathbb{Q}\cap [0,1]$ has zero measure using this definition, and as a result the Dirichlet function has zero integral on the unit interval, while it is impossible to define the Riemann integral rigorously. However, I noticed that by this point he was more or less lost when I showed $m(\mathbb{Q}\cap [0,1])=0$. I told him that to understand it properly he needed to take a year of real analysis classes, and that he should consult a professor I know in my department.
I want to ask what is a good way to explain the ideas of the Lebesgue integral to a student like him next time without making him/her confused. I later learned that the student did not have a proper proof writing background (like he did not know what it means to be injective and surjective). Since these students constitute the majority of my my classes, I feel obliged to find a way to explain myself better without forcing them to pay a visit to my professor or read a serious textbook.