I am trying to understand parts of the behaviour of functions in $L^1(0,T;L^1(0;1))$. More precisely the question I want to answer is the following:
Let $\omega\subset (0;T)\times (0;1)$ be a measurable subset with positive volume. For a.e. $t\in (0;T)$ we define $\omega_t:=\omega\cap(\{t\}\times \Omega)$. Let $X$ be the set of functions $f\in L^1(0,T;L^1(0;1))$ supported in $\omega$. Is it true that for a.e. $t$ such that $|\omega_t|>0$, for any $h\in L^1(\omega_t)$, there exists $\tilde h\in X$ such that in the sense of weak convergence of measures the following convergence holds: $$ h=\lim_{\epsilon\to 0}\frac1{2\epsilon}\int_{t-\epsilon}^{t+\epsilon}h(s,\cdot)ds? $$
Essentially, I am trying to understand if the well-known extension theorems that are valid in the open/continuous setting still hold in the measurable/Lebesgue integrable context.