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What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively.

What I mean is can $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)$ be written as some n-dimensional interval $(..)\times.....\times(..)$ where in the bracket there would be normal set operations $\cup \ \text{or}\ \cap $

with this respect, I am trying to solve: What is the measure of $A=[-1,2]\times[0,3]\times[-2,4]\cup[0,2]\times[1,4]\times[-1,4] \setminus [-1,1]^3$?

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  • $\begingroup$ Think in 2 or 3 dimensions: $A_1\times A_2$ is a rectangle in the plane. Then, for the union, you should take the intersection -if not empty- and split into more rectangles along the lines. If the intersection is empty, the measures are simply added. $\endgroup$
    – Berci
    Commented Jun 12, 2016 at 13:43

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You can find the measure of $A\cup B$ by application of: $$\mu(A\cup B)=\mu(A)+\mu(B)-\mu(A\cap B)$$

The union is in general not a product of intervals here, but the intersection is.

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