Integrals Over More General Sets
To this point we’ve only discussed multiple integrals over -dimensional intervals. But often we’re interested in more general regions, like circles or ellipsoids or even more rectangular solids that are just tilted with respect to the coordinate axes. How can we handle integrating over these more general sets?
One attempt might be to fill up the region from inside. We can chop up regions into smaller pieces, each of which is an -dimensional interval. But overall this requires an incredibly involved limiting process, and we’ll never get anything calculated that way.
Instead, we come at it from outside. Given a bounded region , we can put it into an -dimensional interval . Then if is a function, we can try to define some sort of integral of over in terms of its integral over .
The obvious problem with is that it includes all the points in that aren’t in , and we don’t want to include the integral of over that region. Worse, what if has a big cluster of discontinuities within but outside of ? Clearly that shouldn’t make fail to be integrable over . What we need is to mask off , like a stencil or masking tape does when painting.
The mask we’ll use is the characteristic function of , which I’ve mentioned before. I’ll actually go more deeply into them in a bit, but for now we’ll recall that the characteristic function of a set is written , and it’s defined as
Now look what happens when we multiply our function by this mask:
Now our function has been redefined outside by setting it equal to zero there. Then we proceed to define
The integral over should be the “integral” over plus the “integral” over the region outside . I put these in quotes because we haven’t really defined what these integrals mean (that’s what we’re trying to do!), but we can make reasonable assertions about properties that they should have, whatever they are.
Now is zero on the outer region, so the second integral is zero. And is just equal to inside , so when “integrating” over itself we may as well just drop the factor. This justifies the definition of integrating over .
It should be clear that this definition doesn’t depend on the interval at all. Indeed, if we have two different intervals and which both contain , then any region the one contains which the other does not must fall outside of , and will be zero there anyway, and so will the difference between their integrals.
Now, this definition isn’t without its problems. The clearest of which is that we’ve almost certainly introduced new discontinuities. All around the boundary of , if wasn’t already zero it suddenly and discontinuously becomes zero when we add our mask. This could cause trouble when trying to integrate the masked function over . We must ask that be Jordan measurable, because this will happen if and only if the boundary has zero Jordan content, and thus zero outer Lebesgue measure. Since the collection of new discontinuities must be contained in this boundary, it will also have measure zero.
This leads us to an integrability criterion over a Jordan measurable set : will be integrable over if and only if the discontinuities of in form a set of measure zero. Indeed, Lebesgue’s condition tells us that the discontinuities of must have measure zero. These discontinuities either come from those of inside , or from the characteristic function at the boundary . By assuming that is Jordan measurable, we force the second kind to have measure zero, and so the total collection of discontinuities will have measure zero and satisfy Lebesgue’s condition if and only if the discontinuities of inside do.