# Approximating/ Slacking up measurable functions

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?

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.
• You can certainly define $g(x,y)$ as the difference of your two $f$'s and use the integral of $g$ as the distance. – Ross Millikan Oct 28 '18 at 17:55