Let $d \in \mathbb N^\*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R\_{\ge 0}), t \mapsto F\_t
$$
be measurable. I would like to ask if there is a measurable function $G:[0, T] \times \mathbb R^d \to \mathbb R\_{\ge 0}$ such that


* $G(t, \cdot) \in L^p (\mathbb R^d; \mathbb R\_{\ge 0})$ for all $t \in [0, T]$.
* $\|G(t, \cdot) - F\_t\|\_{L^p} = 0$ for a.e. $t \in [0, T]$.


Thank you so much for your elaboration!



Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?
$\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\LL}{\mathcal L}\newcommand{\si}{\sigma}$The answer is yes, at least for $p\in[1,\infty)$.


Indeed, $L^p(\R^d)$ is a separable metric space. So, for each real $\ep$ there is a countable measurable partition $(B\_{\ep,j})$ of $L^p(\R^d)$ such that for each $j$ we have $B\_{\ep,j}\ne\emptyset$ and the diameter of $B\_{\ep,j}$ is $\le\ep$. Pick any $y\_{\ep,j}$ in $B\_{\ep,j}$. For $(t,x)\in[0,T]\times\R^d$, let
\begin{equation}
 G\_\ep(t,x):=\sum\_j y\_{\ep,j}(x)\,1(F(t)\in B\_{\ep,j}),
\end{equation}
where $F(t):=F\_t$. Then for any real $c$
\begin{equation}
 \{(t,x)\in[0,T]\times\R^d\colon G\_\ep(t,x)>c\}
 =\bigcup\_j F^{-1}(B\_{\ep,j})\times y\_{\ep,j}^{-1}((c,\infty))
 \in\LL([0,T])\otimes\LL(\R^d),
\end{equation}
where $\LL(\cdot)$ denotes the Lebesgue $\si$-algebra. So, the function $G\_\ep$ is measurable, for each $\ep$.


Also, $\|G\_\ep(t,\cdot)-F(t)\|\_{L^p(\R^d)}\le\ep$ and hence $\|G\_\ep(t,\cdot)\|\_{L^p(\R^d)}\le\ep+\|F(t)\|\_{L^p(\R^d)}$ for each $t\in[0,T]$.


Since $F\colon[0,T]\to L^p(\R^d)$ is measurable and the norm on $L^p(\R^d)$ is continuous and hence measurable, we see that the function $[0,T]\ni t\mapsto w(t):=\dfrac1{1+\|F(t)\|^p\_{L^p(\R^d)}}\in[0,\infty)$ is measurable. So, for each real $\ep>0$ we have $G\_\ep\in L^p\_w([0,T]\times\R^d)$, where $L^p\_w([0,T]\times\R^d)$ is the space of all measurable functions $H\colon[0,T]\times\R^d\to\R$ with norm
$$\|H\|\_{L^p\_w([0,T]\times\R^d)}:=\Big(\int\_0^T dt\,w(t)\,\|H(t,\cdot)\|\_{L^p(\R^d)}^p\Big)^{1/p}<\infty.$$


For all integers $m,n$ such that $m\ge n\ge1$
\begin{equation}
 \|G\_{1/m}-G\_{1/n}\|\_{L^p\_w([0,T]\times\R^d)}^p
 =\int\_0^T dt\,w(t)\|G\_{1/m}(t,\cdot)-G\_{1/n}(t,\cdot)\|\_{L^p(\R^d)}^p\le(2/n)^pT\to0 
\end{equation}
as $n\to\infty$. So, by the completeness of $L^p\_w([0,T]\times\R^d)$, for some sequence $(n\_k)$ of natural numbers going to $\infty$ there is a limit $G$ of $G\_{1/n\_k}$ in $L^p\_w([0,T]\times\R^d)$. Clearly, this limit $G$ satisfies your desired conditions. $\quad\Box$


