If $f_1, f_2, f_3,...$ are $M$-measurable, prove that $\sup_k f_k, \inf_k f_k, \lim \sup_k f_k, \lim \inf_k f_k, \lim_k f_k$ (if it exists) are all M-measurable.
My thoughts: We know for any sequence $A_n$, $\inf A_n\le\lim\inf A_n\le\lim\sup A_n\le\sup A_n$. But how do we define this for function $f_k$? I think we'll just define it pointwise for each $x$ in the domain of $f_k$. The same inequality should also be true for sequence of functions.