I am reading a paper.
They define $ L_{p,q}(Q) = \{ u \in L_p((0,T); L_q(Y)) : u(t, \cdot) = 0 \text{ on } Y \backslash Y_t \text{ for a.e. $t \in (0,T)$}\}$ with norm $\lVert u \rVert_{L_{p,q}(Q)} = \left(\int_0^T \lVert u(x) \rVert^p_{L_q(Y_t)} dx\right)^{1\over p}$ for $p < \infty.$
They write
Since $q < \infty$, the function $x \mapsto \lVert u(x, \cdot) \rVert_{L_q(Y_t)}$ is measurable by Fubini's theorem. Thus, the space $L_{p,q}(Q)$ is well-defined.
Can someone explain this to me? By well-defined I guess they want to show that that norm inside the integral in the norm of the space $L_{p,q}(Q)$ exists. Is that right? I don't see how Fubini's theorem tells us that that function is measureable. And I guess measurability implies that it can be integrated over $(0,T)$ like in the norm?