# Example of a measurable function

Let $$f:\mathbb{R}^{2n}\times\mathbb{R}\to\mathbb{R}$$ be a measurable function and $$u:\mathbb{R}^n\to\mathbb{R}$$ be a given measurable function such that $$f(x,y,u)$$ satisfies $$C_1(u(x)-u(y))\leq f(x,y,u)\leq C_2(u(x)-u(y))$$ for every $$x,y\in\mathbb{R}^n$$, for some constant $$C_1,C_2>0$$

I can see that $$f(x,y,u)=u(x)-u(y)$$ itself satisfies the above hypothesis with $$C_1=C_2=1$$. Can someone please give some nontrivial example that satisfies the above hypothesis on $$f$$.

Thank you very much.

• Is $u$ a function or a real number? Commented Sep 10, 2021 at 14:18
• Thanks for the question. u is a function. I edited it. Please see it. Commented Sep 10, 2021 at 14:25
• What do yo mean by $f(x,y,u)$ if $u$ is a function?
– user592521
Commented Sep 10, 2021 at 14:26
• Hi, $f$ is a function of $u$. That's why $f$ is written like this. Commented Sep 10, 2021 at 14:34

First remark, if $$u$$ is a function, either you can write for example $$f(x,y,u(x))$$ or you should have defined $$f : \mathbb R^{2n}× F(\mathbb R^{n},\mathbb R)\to \mathbb R$$ where $$F(\mathbb R^{n},\mathbb R)$$ is the set of functions from $$\mathbb R^{n}$$ to $$\mathbb R$$. Let take this last convention.
To have such a function $$f$$, you can take any function $$g : \mathbb R^{2n}× F(\mathbb R^{n},\mathbb R)\to \mathbb R$$ globally bounded by above and below (if you need explicit examples, take for example $$g(x,y,u) = \sin(x)+\sin(y)\sin(u(x))$$), then define $$f(x,y,u) = g(x,y,u)\,(u(x)-u(y))$$
And actually, the converse is true! If $$f$$ is a function verifying your assumptions, then you can define $$g(x,y,u) = \frac{f(x,y,u)}{u(x)-u(y)}\,\mathbf 1_{\{u(x)-u(y)≠0\}}$$ which is a bounded function of $$x$$, $$y$$ and $$u$$.
• I wonder in such a case, if the function $f$ should be defined as $f(x,y,u(x),u(y),u(x)-u(y))$, since $u$ can take values at $y$ and if we assume $f$ is measurable in all the variables. Commented Sep 12, 2021 at 14:23
• Basically, I am unable to make it clear what should be the domain of the function measurable function $f$ which satisfies the given condition $C_1(u(x)-u(y))\leq f(x,y,u)\leq C_2(u(x)-u(y))$ for every $x,y,\in\mathbb{R}$. From here, it seems to me the points in domain should be $(x,y,u(x),u(y),u(x)-u(y))$. Please have a look. Thanks. Commented Sep 12, 2021 at 15:06