I just read in some comments, that some children got measles one week after getting the MMR shot. Afaik. MMR contains attenuated live virus, and the incubation period of measles is 10 days (+/- ? days). Can we be certain, that they didn't get infected by the vaccine?
There are two possibilities for getting the measles directly after a vaccination:
The kids where infected shortly before the immunization. Since it takes about two weeks to develop humoral immunity, it is possible that these two events overlap. The measles have an incubation period of 7 - 21 days (see reference 1), with a rash occuring around day 14. So if the kid was around day 7 of his infection, this would fit.
Since the measles vaccination is a live attenuated virus, it is possible to get a rash from it. However, these are relatively rare cases (depending on the source between 2 and 5%) and the infection is mild and not contagious. See also reference 2.