Step 1: Understanding the Concept
When a person recovers from measles, the body has met the virus and made its own antibodies and memory cells. This is active immunity.
Step 2: Natural or artificial
The antigen exposure came from a natural infection, not from a vaccine. So it is naturally acquired.
Step 3: Why not passive
Passive immunity means ready-made antibodies and fades quickly. Memory cells formed here give protection for the whole life.
Final Answer:
The person develops natural acquired active immunity.
\[ \boxed{\text{(C) }\text{natural acquired active immunity}} \]