...just carefully physically define the wave function to see why...
Wikipedia describes the (non-existent) measurement problem as follows:
The measurement problem in quantum mechanics is the problem of how (or whether) wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.These sentences are very representative of the confusion of the real-world people who repeat the totally incorrect statement that there is something wrong, illogical, inconsistent, incomplete, or scientifically unsatisfactory about the basic postulates of quantum mechanics.
OK, so the first sentence conveys the message that these people don't understand "the collapse of the wave function". On the other hand, the second sentence pretty much answers the would-be question in the first question – the collapse cannot be observed, at least not in detail, as a sequence of steps – and it postulates that people must be split to many "interpretations", like in a multicultural society, and answer all the questions differently depending on their camp.