Re: Integral help?


Robert Israel wrote:
In article <1162348018.792066.47050@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<adomplayer@xxxxxxxxx> wrote:
Let f:[0,1]x[0,1] be defined by f(x,y) = (x-1/2)^(-3) if y < |x-1/2|,
f(x,y) = 0 otherwise. What can we say about int_0^1 int_0^1 f(x,y)
dxdy, int_0^1 int_0^1 f(x,y)dydx, and int_E f(x,y) (dy X dx) where
E=[0,1]^2 and (dy X dx) is the product measure?

OK, so what do you get when you do those iterated integrals?

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

If you do them in one order, the triangle to the left of x=1/2
contributes -infty and the triangle to the right of x=1/2 contributes
+infty, so I *think* it is rigorous to say the total volume then is
undefined?
If you do them in another order, each horizontal slab contributes 0,
making the total volume 0.
But as for the product measure, I have no idea how to compute product
measures when Tonelli-Fubini fail. I mean, Fubini is basically the
canonical way to compute product measures, to the point where it's very
often used as the *definition* of product measures, in basic calculus
courses. It would be very helpful if you could give me some pointers
how to make sense of/compute the integral of the function wrt the
product measure, for the above function.

Thank you for helping :)

.