Could my child have measles if he broke out in a rash after getting the MMR vaccine?

NO. The rash is a "normal" reaction occurring in maybe 1-2% of the vaccination given. It is not the disease.
No. The rash a week or more after the MMR actually represents a confirmation that the vaccine produced a good response. The vaccine measles component is a weakened cousin of the wild measles. It spends a while challenging babies immune system followed sometimes by a rash. This represents new antibodies that have trapped the vaccine virus and carried them to the skin. It will fade and is not a threat.
No. Approximately 5% of children develop a rash a few weeks after the MMR vaccine. This is considered normal and is not an allergic or adverse reaction. In this day and age of people not vaccinating their children, there is a small possibility your child was exposed just before receiving the vaccine. However if the rash indicates measles, your child would appear ill and must be examined.