Convergence in $L_1$ and Convergence of the Integrals

Question

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 $L_1(a,b)$?

Answer 1

$L^p(a,b)\subset L^1(a,b)$, $p\geq1$ and $a,b\in\mathbb{R}$. Then you have the same result.

Answer 2

Actually, it's neither specific to $\mathbb L^1$ nor to the involved measure space. Indeed, for each $1\leqslant p\lt \infty$, we define a norm on $\mathbb L^p$ by $$\lVert f\rVert_p=\left(\int_X|f(x)|^p\mathrm d\mu\right)^{1/p}.$$

And using triangular inequality, we have that $\lVert f_n-f\rVert_p\to 0$ implies $\lVert f_n\rVert_p\to \lVert f\rVert_p$.