On page 263 of his book Stromberg gives the following definition ($M_0$ denotes the set of real-valued step functions on $\mathbb{R}$).
What is the exact meaning of $\phi_n(x)\to f(x)$ a.e.? Is it $\lambda(E^{c})=0$ where $\lambda$ is Lebesgue measure and $E=\{x\in \mathbb{R} : f(x) \text{ is defined and } \phi_n(x)\ \to f(x)\}$?
Stromberg also makes the following remark.
But this means that $f(x)$ could take value $\pm\infty$ on a set of measure zero and still be in $M_1$. Hence in definition (6.10) it seems like $f(x)$ does not need to be strictly real-valued, but only real-valued almost everywhere.
Am I understanding this correctly? Is it better to view $f$ as a function defined on all of $\mathbb{R}$ but whose values are allowed to be arbitrary on a set of measure zero?
Thanks a lot for help.
Best Answer
About your first point: it should mean that there exists a set $E\subseteq \mbox{Domain}(f)$ such that $\lambda(\mathbb R - E) = 0$ and for all $x \in E:\ \phi_n(x)\to f(x)$.
On your second point: if you have a function $f$ that has infinities only in a set of measure zero, you can still consider the restricted function $f' = f|_{f^{-1}(\mathbb R)}$, and ask if $f'$ is in $M_1$. The functions $f$ and $f'$ are equal a.e.. The assumption in the definition is not a real restriction since for anything that matters you will always work in spaces where a "function" is really an equivalence class of functions that are equal a.e.