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Suppose $f_1(x,y)$ and $f_2(x,y)$ are two bounded real-valued measurable functions on $\mathbb{R}^2$.

Suppose $f_1(x,y) \geq f_2(x,y) \forall (x,y) \in \mathbb{R}^2$.

Can we write $f_1(x,y) = g(x) + f_2(x,y)$?, where $g(x)$ is some bounded real-valued measurable function on $\mathbb{R}$.

In other words, can a single variable function make up for the slack.?

If not how to define closeness and approximate?

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Normally not. This requires that $f_2(x,y)-f_1(x,y)$ be a function of $x$ only. As an example, let $f_1(x,y)=0, f_2(x,y)=e^{-(x^2+y^2)}$. Because $f_2$ depends on $y$ you can't.

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  • $\begingroup$ I see. How would one define closeness with measurable functions? $\endgroup$ – matzgud_89 Oct 28 '18 at 17:47
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    $\begingroup$ You can certainly define $g(x,y)$ as the difference of your two $f$'s and use the integral of $g$ as the distance. $\endgroup$ – Ross Millikan Oct 28 '18 at 17:55

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