# Integral over a bounded set

I don't fully understand this theorem:

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?

mathman
f is 0 outside of the intersection. Therefore the integral of f is determined by its value in the intersection. Q = Q∩Q' + Q-Q' and Q'=Q∩Q' + Q'-Q. Since f = 0 on both Q-Q' and Q'-Q, the integral of f is the integral over Q∩Q', which means the integral over Q and the integral over Q' are the same.

Why does it say then that "then f vanishes at SOME point in R" and not rather in ALL of R?

Using what it says it says that it follows that m_R(f) is less than or equal to 0. But if f(x) were equal to 0 on all of R which it seems like it should be (since R isn't in the intersection but rather only in Q'), then m_R (f) shouldn't be less than zero.

It goes on to say that:

L(f,P") less than or equal to sum of [m_R(f)]vol(R), where R are the sub rectangles contained in Q and m_R (f) = inf{f(x)| x in R}.

Thanks for your help, mathman I appreciate it.

Office_Shredder
Staff Emeritus
Gold Member
If R is not a subset of the intersection, then some points of R are in the intersection (maybe none of them but we don't know), and some points are outside the intersection

Ahh yes that's true. I guess the only R which may have m_R (f) < 0 are the ones which are along the boundary of Q as they have some points of Q inside. If they don't, however, m_R (f) should be strictly equal to 0.

Thanks a lot for your help Office Shredder! :)

mathman
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'.

That was the part of your comment I was trying to address. Your further comment seems to be a distortion of the original problem.

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?
This statement is completely wrong! The original statement is the function is 0 OUTSIDE the intersection, and anything INSIDE!