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That is my problem... $f$ is decreasing, continuous, $f:[0,1]^2\to[0,1]$, $f(x,y)=f(y,x)$ [Symmetric function] and I have some marginal conditions like $f(x,1)=f(1,x)=0$.

Is it possible to compute the double integral $\int_{0}^1 \int_{0}^1 f(x,y)dxdy$ using some marginal integration?

For example, when is this true? $$\int_{0}^1 \int_{0}^1 f(x,y)dxdy= \left[\int_{0}^1 f(z,a)dz\right]^2,$$ for some value $a \in [0,1]$.

ex: $f(x,y)=(1-x)(1-y)$ my statement is true, using $a=0$.

Thanks!!

TMM
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    What do you understand by decreasing for a multivariable function? – TZakrevskiy Mar 03 '14 at 22:04
  • Hi TZ, Thanks for asking! In my problem,it is just marginal decreasing... I mean, for each y* (fixed), f(x,y*) is decreasing in x. Also this happens for other coordinate. Thanks! D – doubt Mar 04 '14 at 14:25

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