It seems opinion is quite divided between checking behind turn, and betting somewhere between $40 and $70 for value/to induce spaz.
Now, onto the next part...
V limps UTG, folded to hero in MP with A
A
who raises to $15, folded to V who calls.
Flop ($34): Q
2
3
V checks, hero bets $20, V tanks then check-raises to $40, hero calls.
Turn ($114): A
V checks, hero checks.
River ($114): 8
V quickly checks, hero bets...
My reasoning for checking behind the turn:
After V min check/raised flop but failed to barrel turn, I believed that he almost never had air. This is because V seemed aware of pot size, and as such would rarely be expecting me to fold on the flop to his bet, even with A-hi. It seemed he had any sort of hand from low-medium pairs to Qx all the way up to 22/33 (I didn't consider QQ in his range).
-On this specific turn, I expected V to be folding his Qx or lower hands to most turn bets.
-Additionally, given his river action in hand 1, if he was attempting to check/raise turn with a set/2P, I would expect him to bomb river, allowing us to raise AI regardless.
-From hand 2 I deduced that there was also a non-zero chance that he might turn a low pair into a bluff on the river, thinking we have a 99-JJ type hand.
-From hand 2, I also believed that if we checked behind a street he would think we're FOS and as such be more likely to pay off a river bet if we check the turn.
-I largely discounted 45 from his range because of how passively he played his draw in hand 1.
Now, on the river it's a simple sizing question.