"The pointless limits of measurable functions is measurable" also struggling with this, do they mean limit in topological sense? it seems like theres also a limit with respect to ordering, like if liminf and limsup are equal?

"The supremum of random variables is random variable" I think this holds as long as they are valued in a measurable space whose sigma algebra is generated by down-sets in a complete lattice (aka intervals [bot, x] where bot is the bottom of the lattice, e.g. negative infinity in the extended reals)