Maybe this is a simple question, but any help would be appreciated!

I've seen both in this book and other places that the risk neutral measure (with respect to a numeraire) is defined as an equivalent measure (to the real world one) such that ALL tradeable assets (discounted by the numeraire) are martingales. Typically then one finds the measure which makes for example the stock/bond (or the bond/stock in the stock numeraire) a martingale and then says that the option/bond (option/stock) is also a martingale and prices this way.

Why does finding an equivalent martingale measure for only one process imply ALL other tradeable discounted processes are martingales with respect to this measure?!

If we have abstractly the existence of a risk neutral measure in the above sense (first fundamental theorem of asset pricing?), i.e. one that makes all discounted tradeables martingales, and if we're in a complete market, then by uniqueness finding the equivalent martingale measure for one process (like stock/bond) is sufficient to know that also the option/bond process is a martingale under this measure.

However, it seems like the same tactic is used in the incomplete models too. For example in the jump diffusion chapter, the stochastic process for the stock is written down (with the Poisson jumps), Girsanov's theorem and the martingale classification theorem are invoked to find an equivalent martingale measure for the stock/bond process, but then the option/bond process is also assumed to be a martingale in this measure and priced via discounted expectation! I understand that risk neutral measures are not unique in this setting (i.e. incomplete), why does finding an equivalent martingale measure for the single stock/bond process give us an equivalent martingale measure for the processes of other tradeables? i.e. Why does the equivalent martingale measure for only the stock/bond process work jointly for the stock/bond and option/bond processes?

Thanks for any help and sorry if I'm misunderstanding something fundamental!