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!
Best Answer
$\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\LL}{\mathcal L}\newcommand{\si}{\sigma}$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. $\quad\Box$