Let $\Omega$ be a bounded smooth subset of $\mathbb{R}^n$ and $T>0$ fixed. And let $h$ be a function defined on $\Omega\times [0,T]\times\mathbb{R}$ with values in $\mathbb{R}$.
For almost everywhere $(x,t)\in \Omega\times [0,T]$, we assume that $h(x,t,.)$ is continuous and for all $s\in \mathbb{R}$ we assume that $h(.,.,s)$ belongs to Lebesgue space $L^{1}(\Omega\times [0,T])$.
Can we approximate the function $h$ by a function $h_{n}\in C^{\infty}(\Omega\times [0,T]\times\mathbb{R})$ such that: $h_n \to h$ in $L^{1}(\Omega\times [0,T])$ for any fixed $s$ and uniformly on compact subset of $\mathbb{R}$ for almost everywhere $(x,t)$ ?