[swede554]

Interesting spot. As you questioned, I think the fact that we rivered a set doesn't really increase our showdown value much vs. his range. I'd imagine villain would raise the flop w/ a set given the drawiness (word?) of the flop, or at least bet the turn. I could see him taking this line w/ 44 though , not wanting to scare you away with a turn bet. That's a possibility. So by the turn we're either ahead or behind (obviously), and I don't think the river changed that, hence our showdown value didn't increase much vs. his range (makes sense?)

However, another thing we should consider is what your hands look like as well. The raise pre, followed by a bet-check-bet ace river looks like AK-AQ, so he could be raising A5-A7 for value if he called you w/ that preflop, but I think that's somewhat unlikely.

All-in-all, yeah we have a set, and that's great and all, but if we're not 100% sure how we're going to respond to a raise if we bet this river, I prefer check-calling. I know this might sound alil weak-tightish, but I kind of like giving villain the opportunity to bluff his busted draws or rep the ace here.