With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral)
What I know:
- With the Lebesgue integral we need less requirements to switch integral and limit of a sequence of functions. (monotone convergence, dominated convergence,..)
- We can integrate much more functions, such as the Dirichlet function
- We also can integrate function over sets, which are not real numbers (in comparison to the Riemann integral)
What else can we add ?
There is a certain "guarantee of correctness" built into the requirement that Lebesgue integrals converge absolutely. For example, although $\int_0^\infty \frac{\sin(x)}{x} dx$ exists in the improper Riemann sense, the value depends on calculating the integral on $[0,b]$ and then sending $b \to \infty$. You get a different value if you take the "positive parts" and add them "much earlier" in the sequence than the "negative parts". Just like with absolutely convergent infinite series, this phenomenon cannot happen with Lebesgue integrable functions.
Your first and second points are closely related, and between them there are a lot of relevant issues. For example, the Lebesgue $L^p$ spaces are complete. Their Riemann counterparts are not complete. (Here I am talking about $1 \leq p < \infty$.) This fact is necessary for various extensions, such as the notion of Sobolev spaces, which are widely used in theoretical PDE.