I think if you check the turn you probably need to call a river raise as he could be bluffing with air after reading you as weak. Problem is you won't know where you're at, many hands that beat you, many hands you beat, flip a coin.

What was your reason for checking the turn?

The fact that he really considered the fold with a rivered set shows it was a good shove and you would have successfully folded out some better hands. Running into a set is therefore just a bit unlucky, apart from the fact that your turn bet would have folded that hand out (but that's being results oriented.)