Show that $\displaystyle\int _{(0,1) \times (0,1)} \frac {xy } {(x^2+y^2)^2 } d(\mu \times \lambda )$ does not exist?

Question

I want to show that $$\int _{(0,1)\times (0,1) } \frac {xy } {(x^2+y^2)^2 } d(\mu \times \lambda )$$ doesn't exist, but don't know how to do it.

First thing would be to try to show that the the iterated aren't equal.But in this case they are, since in any order, they both equals zero.

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Does $f \in L^1 (\mu \times \lambda ) $ imply $$\int d \mu\int \left|\frac {xy } {(x^2+y^2)^2 } \right| d \lambda < \infty$$ ?

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I know $$\int d \mu\int \left| \frac {xy } {(x^2+y^2)^2 } \right| d \lambda < \infty$$ is a suficient condition for $f \in L^1 (\mu \times \lambda ) $ but is it also necessary?

[I can show that $\int d \mu\int |\frac {xy } {(x^2+y^2)^2 } | d \lambda = \infty$]

If this is not the case, then what?

Thanks in advance!

Answer 1

Hint: make the polar coordinates change of variables.

You will get something of the form $$ \int_{[0,\pi/2)} d\theta \int_{(0,\epsilon)} dr \frac{F(\theta)}{r^2} + O(1) $$