Is it possible to construct a non-negative Lebesgue measurable function $u:{\bf R}\rightarrow {\bf R}$ with the following property:
For every sequence of Lebesgue measurable sets $(\Omega_n)$ in ${\bf R}$ such that $\lim_{n\rightarrow+\infty}\lambda({\bf R}\backslash\Omega_n)=0$ and every $n_0\in {\bf N}$ there exists $n\geq n_0$ (which depends on the sequence $(\Omega_n)$) and $M_n\geq 0$ (which also depends on the sequence $(\Omega_n)$) such that $u(\Omega_n)=[M_n,+\infty)$?
DISCUSSION.
Remark0. The point here is that the aforementioned property holds for every sequence of Lebesgue measurable sets $(\Omega_n)$ as above. Such a requirement makes the construction of the measurable function $u$ difficult (at least for me it appears to be difficult; I was not able to construct such a measurable function $u$).
Remark1. I do not know if measurability of $u$ is an essential condition here. To begin with, the construction of any function (not necessarily measurable) with the aforementioned property is also of interest.
Remark2. I believe that there is a connection with the Luzin Theorem here (see
Significance of Luzin's theorem.
). Namely, I think that, if the domain of the function $u$ is a set of finite measure $\Omega$ (instead of ${\bf R}$), by the Luzin Theorem, it results that for every $0<\eta<1$ $u$ is continuous on some compact sets $K_{\eta}\subseteq\Omega$, where $\lambda(\Omega\backslash K_{\eta})\leq\eta$, and therefore $u$ is bounded on each $K_{\eta}$, so $u:K_{\eta}\rightarrow {\bf R}$ can not be surjection on $[M,+\infty)$ for any choice of constant $M\geq 0$. So the function $u:\Omega\rightarrow {\bf R}$ with the aforementioned surjectivity property does not exist.
Remark3. $u(s):=s$ is not an example, since we can choose a sequence of sets $\Omega_n$ such that $\Omega_n\cap [M,+\infty)$ is non-empty for every $M>0$ and every $n\in {\bf N}$ and such that $\lim_{n\rightarrow+\infty}\lambda({\bf R}\backslash\Omega_n)=0$.It results that $u(\Omega_n)$ is not equal to $[M,+\infty)$. From this we can conclude that, if $u$ has the aforementioned property, then $u$ has much stronger property than that $u$ is a mere surjection on $[0,+\infty)$, and $u$ is probably not an injection. My view is that rapidly oscillatory functions are probably the class of functions which could satisfy the desired property.
Remark4. There is probably a connection between this question and the following discussion, involving the so-called "Strong Darboux functions" ( see book Ciesielski: Set Theory for the Working Mathematician p.106):
