After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these exercises.
Suppose $\theta_n$ and $\phi_n$ are mean consistent estimators for $\theta,\phi$.
Prove: $\theta_n +\phi_n\rightarrow^{\mathcal{L}_1}\theta +\phi$ and $\max(\theta_n,\phi_n)\rightarrow^{\mathcal{L}_1}\max(\theta,\phi)$
Thus far I have gathered/surmised that the definition of mean consistent estimators essentially says:
$\theta_n\rightarrow^{\mathcal{L}_1}\theta $
$\phi_n\rightarrow^{\mathcal{L}_1}\phi$
Even under the assumption that $\theta_n\rightarrow\theta$ a.s, I am somewhat confused on how to proceed.
Integrate the inequalities $$ |(\theta_n+\phi_n)-(\theta+\phi)|\leqslant|\theta_n-\theta|+|\phi_n-\phi|, $$ and $$ |\max\{\theta_n,\phi_n\}-\max\{\theta,\phi\}|\leqslant|\theta_n-\theta|+|\phi_n-\phi|. $$