In probability theory, I have often come across the identification $$ L^p_{\mathbb{P}}({\mathcal{F}})\otimes L^p_{\mathbb{P}}({\mathcal{F}}) \cong L^p_{\mathbb{P}\otimes \mathbb{P}}({\mathcal{F}\times \mathcal{F}}), $$ where $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space and $\mathbb{P}\otimes \mathbb{P}$ denotes the product measure on the product $\sigma$-algebra $\mathcal{F}\otimes \mathcal{F}$.
However, I have recently learned that there are many topologies on these tensor products; each with different properties? So which topology is the above identification valid under?
For example, is it the projective topology, the injective topology, or the so-called "a-topology" which is coarser than the two above and is characterized by if $T_1$ and $T_2$ are continuous linear operators on $L^p_{\mathbb{P}}{\mathcal{F}} $ then $T_1\otimes_a T_2$ is a continuous linear operator on $L^p_{\mathbb{P}}({\mathcal{F}}) \otimes_a L^p_{\mathbb{P}}({\mathcal{F}} )$.
Thanks in advance for helping; this is very far from my field.