Is this a measurable function is this space?

Question

Let $\Omega \subset R^n$ ($n\geq 2$) a bounded domain. Let $u \in L^{1}(\Omega)$. Let $F(x,k):= \chi_{\{ u> k\}}(x), (x,k) \in \Omega \times (-\infty , + \infty) $ . This function is measurable and integrable with respect to the product space $\Omega \times (-\infty , + \infty)$ ? I am asking by curiosity.. I dont know how to prove this ..

Answer 1

$F$ is either $0$ or $1$, and we have

$$F^{-1}(1)=\{(x,k):u(x)>k\}.$$

This set is measurable as $u$ is measurable (since it is integrable). Moreover, $F^{-1}(0)$ is the complement of this set.

$F$ does not have to be integrable however. Let $u\in L^1(\Omega)$ such that $u$ is a positive function (for example a Gaussian). Then:

$$\int F(x,k)dxdk=\int \chi_{u>k}(x)dxdk=\int \mu(\{u>k\})dk,$$

where $\mu$ denotes the Lebesgue measure. Let $\mu(\{u>0\})=s$, then $s$ is a positive number, and $$\int \mu(\{u>k\})dk\ge \int_{-\infty}^1\mu(\{u>k\})dk=\sum_{n=-\infty}^0\int_n^{n+1}\mu(\{u>n\})dn\ge\sum_{n=-\infty}^0s=\infty.$$

This is true for general (i.e. bounded or unbounded) $\Omega$.