Consider vector bundles on any smooth projective variety $X$. It is easy to see $\mu$-stablility always implies indecomposability. On the contrary, I read the following argument for $X=\mathbb P^1$:
Let $E$ be an indecomposable vector bundle on $\mathbb P^1$. By Harder-Narasimhan filtration and Serre duality, $E$ is $\mu$-stable.
I have difficulty in following this. We need to show for any subbundle $F\subset E$, it always satisfies $\mu(F)\leq \mu(E)$. This will imply that the Harder-Narasimhan filtration will necessarily to be trivial, but I don't know if this is sufficient. And I also don't know how to prove this. (Precisely I don't know how to use Serre duality here.)
And please do not use the fact that $E$ actually splits. (This is on the way to prove it)
Thanks in advance!