Conditional expectation process is a martingale

By conditional existence the set of real measurable integrable versions at every time is nonempty. AC selects one member for each $n\in {\mathbb{N}}_0$. Thus each $M_n$ is ${\mathcal{F}}_n$-measurable and integrable, so $M$ is an integrable adapted process. AC also covers the RN existence assumption.

Since ${\mathcal{F}}_n\subseteq {\mathcal{F}}_{n+1}$, the tower identity gives $E[M_{n+1}\mid {\mathcal{F}}_n]=E[E[X\mid {\mathcal{F}}_{n+1}]\mid {\mathcal{F}}_n]=E[X\mid {\mathcal{F}}_n]=M_n$ a.s. for each $n\ge 0$. This is the martingale condition Martingale submartingale and supermartingale. If representatives of these identities are specified, their measurable failure sets have probability zero; their countable union is measurable and null. This does not complete any ${\mathcal{F}}_n$ or alter its representatives.