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Let Q and Q' be two rectangles in R^n. If F: R^n -> R is a bounded function that vanishes outside Q intersect Q', then integral of f over q is equal to the integral of f over Q'.

When it says that the function vanishes outside of Q intersect Q', does it mean its always zero in the intersection, or simply at some points? See if it only vanishes at some points I don't see how the theorem could be true. The proof considers the special case when Q is a subset of Q' and the partitions Q'. Then it creates a refinement of that partition by adjoining the endpoint of Q into the partition of Q'. However, it goes on to say that if R is a sub rectangle that is not contained in Q then f vanishes at SOME point in R. But I understood that it vanishes in the intersection and thus should completely vanishes on R.

What am I not understanding?