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I am looking for a function $f$ defined over $Q=\left[0,1\right]\times\left[0,1\right]$, with both iterated integrals defined, but $\int_Q f$ does not exist.

I found some examples here, but I am curious if there is any general patterns underlying these functions. I remember that in differential analysis, $$\frac{\partial f(x,y)}{\partial x \partial y}=\frac{\partial f(x,y)}{\partial y \partial x}\iff\text{$f(x,y)$ is $C^2$}.$$ I am looking for similar conditions for iterated integrals to agree, or $$\int_0^1\int_0^1 f\text{ }dx\text{ }dy=\int_0^1\int_0^1 f\text{ }dy\text{ }dx\iff??$$

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Sekots Reivan
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  • Duplicate, but not answered: http://math.stackexchange.com/questions/1480442/example-of-distinctions-between-multiple-integral-and-iterated-integrals – Henricus V. Mar 29 '16 at 15:01
  • @HenryW., our two questions are similar, but that question does not ask for a necessary and sufficient condition for the statement to be valid. – Sekots Reivan Mar 29 '16 at 15:06

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