Parallel to Munger’s “Show me the incentives and I will show you the outcome” which I think all of us have or will realize for ourselves at some point in life.
Strictly speaking this is not the contrapositive and therefore the proof is yet to be seen. A sound corollary: "If I do not behave, it is because you did not measure me."
Yes, a corollary can be just the contrapositive of something you just proved. Sometimes it's even more trivial, like a special case of a general theorem you proved.
A very common use is to re-state something so it's in the exact form of something you said you'd prove. Another common case is to highlight a nice incidental result that's a bit outside the path towards the main result -- for example, it immediately follows (perhaps logically equivalent to) something that's been proven, but it's dressed in a way that catches the attention of someone who's just skimming.