Convergence in $L_1$ and Convergence of the Integrals Am I right with the following argument? (I am a bit confused by all those types of convergence.)
Let $f, f_n \in L_1(a,b)$ with $f_n$ converging to $f$ in $L_1$, meaning
$$\lVert f_n-f  \rVert_1 = \int_a^b |f_n(x)-f(x)|dx \rightarrow 0 \ , $$
Then the integral $\int_a^b f_n dx$ converges to $\int_a^b f dx$. To show this we look at$$\left| \int_a^b f_n(x) dx - \int_a^b f(x) dx \right | \leq \int_a^b | f_n(x) - f(x)| dx \rightarrow 0 \ .$$
If this is indeed true, is there something similar for the other $L_p(a,b)$ spaces, or is this something special to $L_1(a,b)$?

Let $f_n \to f$ in $L^p(\Omega)$. Then, we also have that $\|f_n\|_p \to \|f\|_p$. So "something similar" holds.
As for the convergence of $\int f_n$ to $\int f$, this is generally not guaranteed by $L^p$ convergence, unless the measure of the underlying space is finite (like it is in your example).
In that case we have
$$
\left| \int_{\Omega} f_n(x) dx - \int_{\Omega} f(x)\,dx \right| = \left| \int_{\Omega} (f_n(x) dx - f(x))\,dx \right| \leq \int_{\Omega} | f_n(x) - f(x)|\, dx \leq |\Omega|^{\frac1q} \|fn-f\|_p \rightarrow 0
$$
by Hölder inequality. This is the same inequality that gives you that $L^p(\Omega) \subset L^1(\Omega)$.
