Jeff Green had 43 points, seven rebounds and four blocked shots in the Boston Celtics' 105-103 loss to the Miami Heat on Monday.
"That's sports, that's the nature of the game. Jeff Green is not a LeBron James, not Kevin Durant, not Larry Bird. He's a really good player for us, and a very important player for us. And I think he's getting better for us every time.
"We don't expect Jeff to go out there and get 30, or even 25, every night."
Ainge was asked if he expects Green to be the next Paul Pierce.
"I don't think it's fair to expect from Jeff what we got from Paul, because Paul is arguably the best offensive player we've ever had in their amazing history. So I think it's unfair to put the expectations of being Paul Pierce on him.
"But as an heir apparent, as our starting small forward at some point when Paul no longer plays, I think that's a fair expectation."