Let $E$ be a vector bundle over $S^2$ with inner product, $\pi$ its bundle projection and each fiber has dimension $m$.
Let $\Omega \subset \mathbb{R^n}$. Consider space of all square integrable functions $f$ from $\Omega\times S^2$ to $E$ i.e. $$ \|f\|^2 = \int_{\Omega \times S^2} \| f(x,\omega) \|^2_{\pi(x,\omega)} \, dx \, d\omega < \infty $$ such that $f(x,\omega) \in \pi^{-1}(\omega)$ for all $\omega \in S^2$. Thanks to this condition the space is a vector space. Denote this space $\mathcal{L}^2(\Omega\times S^2,E)$.
Is this space $\mathcal{L}^2(\Omega\times S^2,E)$ any different to the standard vector valued Lebesgue space $L^2(\Omega\times S^2,\mathbb{R}^m)$ ?
My guess is no, because it really does not care about the topology of $E$.
If they are different where I can read more about such space?