For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable wrt. $t$ (the intention is that $\Omega_0$ evolves in a smooth way without jumps).
Define $Q=\bigcup_{t \in (0,T)}\Omega_t\times \{t\}$. We can put the Lebesgue measure on $Q$ if we think of $Q$ as a non cylindrical domain.
Given $u:Q \to \mathbb{R}$, define $\tilde u:\Omega_0 \times(0,T) \to \mathbb{R}$ by $$\tilde u(x,t) = u(F_t(x), t).$$
My question is, given $u \in L^p(Q)$, is $\tilde u \in L^p(\Omega_0 \times (0,T))$?
And conversely, given a function $\tilde u \in L^p(\Omega_0 \times (0,T))$, is $u \in L^p(Q)$, where $u(x,t) :=\tilde u(F_t^{-1}(x),t)$?
The only issue I have is with measurability. Assume that the integrals are all finite. I just need to know whether the functions are measurable.
See this thread for more details about the disjoint union, and this thread for one step of the problem.
Note that I am using the equivalence between $L^p(0,T;\Omega_0)$ and $L^p(0,T;L^p(\Omega_0)$ here (for $p \neq \infty$).