You may be trying to read more into that statement than is intended.
The first requirement of the required algorithm is that the result is a probability as our expectations can only be expressed as a probability.
That is, probabilities can't be greater than one or less than zero; thus one might think that the algorithms available to us are limited in some way. How can you be sure you have included all possibilities in your analysis.
It turns out that any mathematical algorithm can be seen as an operation which transforms a given set of numbers into another set of numbers. Thus psi is now "any" such algorithm and we can be sure no possibility is omitted by our notation: that is, if an algorithm exists which will provide us with what we want, it is a member of the set referred to as "psi" here.
But psi doesn't qualify as it is not constrained to be between zero and one. We can accomplish that requirement by creating psi dagger which is exactly the same as psi except that every number in the set produced by the algorithm is replaced with its complex conjugate. Then we define the dot product as the sum of all those products (each output of psi multiplied by its complex conjugate).
The product thus developed has to be a positive real number. The only problem with this is that the result can be greater than one. That is fixed by the process of "normalization". Just integrate P over all variables. That is the probability of any possibility happening which is one by definition. All we have to do in order to "normalize" psi is replace that psi with an identical algorithm divided by the square root of that number. Now, if we do the integral, we will get one so the product can be interpreted as a probability.
It follows that, if an algorithm exists which will yield our desired probabilities, it can be written with the notation given. We are assured that we have included all possible solutions in the notation.
If that is insufficient, there is a more involved explanation in
http://home.jam.rr.com/dicksfiles/reality/CHAP_I.htm
Scroll down to equation 1.3 and read the section following that expression.
If you have any further questions let me know.
Have fun  Dick
