Measurement in quantum mechanics is something which is very dubious. I reckon that the collapse of wave function itself corresponds to what 'measurement' is. As per wikipedia, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. I think, we can rephrase what measurement is in a less 'handwavy' manner as an act which causes the wave function to collapse (and yields a numerical result). Now, the question of 'what physically constitutes a quantum measurement?' can be rephrased as 'What kind of physical interaction causes the collapse of wave function?'. Simply put, no one knows answer to that yet, and contrary to what a lot of ppl might say, various interpretations don't solve this problem either.