Measurability of $L^{p}(L^{q})$ integrable functions

Question

Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that $ \int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty $ In addition we have that the function given by $ g(x) = \int_{0}^{\infty} F(x,r) h(r) \, dr$ Is measurable for any $h \in L^{q'}(\mathbb{R})$. Then is it true that $F(x,r)$ is Lebesgue measurable ? as a function defined on $\mathbb{R}^{n+1}_{+}$