There is a reason why probabilities are assigned and not measured : it's because probability is a mathematical concept, and in mathematics we do not measure things because there is no such thing as experience in mathematics (at least the theoretical part of it). In probability theory, when one defines a probability over a probability space, it is up to the mathematician to decide what probability is attributed to each event ; whether he sees fit that one probability works better than another is up to him.
Now what happens in real life is that there are standard intuition for probability definitions ; the most basic one would be to define the probability of an event as the number of possible cases where it happens over the number of possible cases in total. That makes sense if you want to count things ; in sampling theory, however, this does not fit the situation. In that case, the sampling plan needs to be adjusted so that the samples are chosen such that every sampling unit has a non-zero probability of being chosen, but there are many different kinds of sampling plans for many different reasons ; the most important one being when a sampling plan needs to take into account auxiliary information to improve accuracy of the sampling. That auxiliary information would've had to be measured, but the sampling plan is an assignment of the probabilities that each sample gets chosen.
I have not studied Bayesian statistics so I'm afraid I can't comment on that.
Hope that helps,