Two dimensional functions: convergence in one dimension

Question

Let $T>0$. Let $f:[0, T]\times \mathbb R \rightarrow [0, \infty[$ such that $f\in L^1([0, T]\times \mathbb R)$. I want to approximate $$\int_\mathbb R f(t,x)dx,$$ by integrals of the form $$\int_\mathbb R g(t,x)dx$$ for $t$-a.e. on $[0, T]$, where $g\in C^\infty([0, T]\times \mathbb R)$.

Is it possible?