Thank you, that was a lovely answer.
I won't be able to fully understand the topic in the link yet, but it's on my to-do list now.
Can you explain a bit on: "a measure gives a value µ(A) to any subset A, and obeys µ(A U B) = µ(A) + µ(B)"?
I understand what subset and union mean, but you didn't say what B is. Also, can you tell me how "obeys µ(A U B) = µ(A) + µ(B)" applies to finding the area of a rectangle?
I hope I'm not being boring by asking these questions and reading more myself. Right now, because I'm short on time, I'm just trying to get a general idea, not delve deeply and look for exact answers.